I think fundamentally the difference between the mindsets of playing a pincer and a check is to do with the difference between sente and the initiative.
When you look at (Japanese) commentaries on old games you see much discussion of which is the right pincer to play, and we can discern that the old players themselves were asking the same question. They experiment with each pincer and if they think they've found a key point, that pincer can become the fashion for a whole generation. But then they end up confused again.
I think the confusion is inherent in the position. A pincer is essentially a tactical move. It is a sente move. It expects a response (the gote move) and then sente follows that gote, another sente/gote follows that, and so on remorselessly. What we fairly typically end up with is a running fight into the centre. Put it another way, we end up with unpredictability. Worse, for the sente player it's very often a predictable unpredictability in that it ends up with having chased his opponent's group into a strong position (eyes or connection) while his chasing stones end up swimming in a sea of weaknesses. He has become the victim of amarigatachi. His opponent has duped him with amashi. Amashi was commoner and/or more obvious in Edo times than in modern times because the rank differences were greater then. Even weakfish players could see the poor results of their play, but didn't have much idea how to improve it. So, try another pincer! What they should have been doing was: try another strategy - maybe a tsume.
Let me go back over some of that paragraph. Nearly everyone reads forum pages far too quickly (the boss might walk in!), and misses the real points. I want to slow things down. My "another strategy."
Sente means making the first stroke as in a sword fight. Gote means making the responding stroke. They go hand in hand, walk in step, cancel each other out, etc. etc. There is thus no inherent real advantage in having sente. We saw that with Muhammad Ali's rope-a-dope against Sonny Liston. Liston took sente, Ali had gote. Liston tried pincering Ali between his massive paws. Ali parried and ran away. Liston ended up a spent force. Ali ended up with the initiative and victory.
A sword fight is mostly about sente and gote. Victory likewise depends on something else: the initiative. Imagine a totally different kind of sword fight. You have a prisoner strapped to a chair. Above him, by a thread, hangs the Sword of Damocles. If he doesn't cooperate, that thread might be snapped. Boy, do you have the initiative - sente is irrelevant.
How might that translate into go? One common way is that you have a sente-gote skirmish, marching step in step, until you decide it is safe to break off and play a move elsewhere where there is a sente but no gote. You have stolen a march. Instead of thinking about keeping on attacking in the skirmish you grab the tedomari (the last available move, where there is a sente but no gote) in another position. This is often described in commentaries by way of phrases such as being the first to get to some move (e.g. a shimari in the last still-open corner). You are now ahead in the balance of territories. You have the initiative. You have real control. You are no longer playing protagonist and deuteragonist in a Greek tragedy. You are now a god on Mt Olympus.
If you are a real god, like an AI bot or Go Seigen, and have Ali's speed of foot, you can even dispense with the skirmish to settle a local position and just tenuki straight into the tedomari.
I believe that many old players understood all that at the results end of the process. They mostly just couldn't see how to get there. Great players like Shuei did see ways, such as proper use of miai. But those insights are what made them great. They were individual, not mainstream.
Eventually, though, the tributary did flow into the main river, and I think there were two main impulses. One was the use of tsume. It might be the same move as a pincer as regards being a point of the board, but the best old players (led by Honinbo Shusai) eschewed the pincer mindset and shifted allegiance to the tsume party. They shifted from sente to initiative.
So why is a tsume initiative-rich rather than sente-rich? I would say the difference lies in the fact that a pincer attacks the opponent whereas a tsume pressurises the opponent. On the surface the difference is actually wafer-thin, and the term 逼 both illustrates and implies that. Japanese pros (amateurs only rarely) use that character in the term 'semaru' = press/approach right up, which - as you can easily see - is related to 'semeru' = attack.
But the difference in mindset can be enormous. With tsume, your subconscious mind is now thnking about safety, prudence, honte, keeping options open (miai), control, the initiative. The pincer player's subconscious is instead looking forward to the roller coaster ride and all the fun of the fair - and it lets your stomach worry about the effects of feeling queasy after guzzling too much candy floss and hot dogs.
A more mundane analogy? You have £100 to spend and walk into an antiques shop. You are the pincer type. You don't want to waste time asking the price of each item one by one, so you say to the dealer: I've got £100 to spend - what can you recommend. He shows you a £5 vase, only it just happens to be on sale now at £100. You buy it. You have been duped. You have been rope-doped. You have suffered amarigatachi. Serves you right. But you learn your lesson and next day you become a tsume player. You do some preparation. You put on a tatty jacket, practise your shrugs in a mirror, and only then go to the shop. You don't tell the dealer anything. You just pick out a piece you like - maybe a £5 vase. You ask what it might cost. Dealer looks at you, decides you could barely afford a coffee, but that vase has been hanging around for quite a while, so he decides to shift it - £5 to you, sir" You delve into your shrug repertoire. The dealer either relents and knocks another couple of quid off the price, or he sticks to his guns. But even then you have lost nothing. You pay £5, not £100.
While I think this process is going on all the time in high-level go, it is not easy to see and even harder to copy - even for other pros. That is why I think another approach to keeping control of the game (i.e. the initiative) was tried.
Everyone who's looked at more than a few Edo games will have noticed an absence of high pincers. What I think was behind the actually rather slow development of high pincer play (I am not counting taisha as a true pincer BTW) was not an attempt to try yet another pincer of the sente-gote tactical type. Rather it was to do with kurai - a position relatively higher than your opponent's. Typically a fourth-line play as opposed to a third-line play. I think Edo players avoided this because the centre was just too hard to control. Too speculative. But modern players had the benefit of the experimentation in the centre of Shin Fuseki and became more comfortable with early centre-facing plays. They also had the benefit of Shuei's games in which he demonstrated control of the game via control of the centre, especially using his favourite L shapes (magaris). Shuei was not quite in the modern category because even he found control of the centre early on a tad too difficult, although with his frequent use of star point corner moves he was moving that way. I think we can safely say that the player who brought the kurai mindset to full fruition was Takagawa Kaku. He referred to it as 'balance'. (And guess who his favourite player was.)
But go is so rich that we could have players like Takagawa's great rival, Sakata Eio, who didn't have much truck with kurai and instead and so often sharpened his razor to cut at the jugular of opposing groups on the second line. Sakata was like the bantam cock in the Corries' famous song:
He was a fine upstanding bantam-cock
So brisk, and stiff, and spry...
With a springy step, and a jaunty plume
And a purposeful look in his eye
In his little black laughing eye!
So I took him to the coop and introduced him to
My seventeen wide-eyed hens
And he tupped and he tupped as a hero tupps
And he bowed to them all, and then
He up and took 'em all again!
Well then upon the peace of my ducks and geese
He boldly did intrude
And with glazed eyes and opened mouths
They bore him with fortitude...
And a little bit of gratitude!
He jumped my giggling guinea-fowl!
He thrust his attentions upon
Me twenty hysterical turkeys,
And a visiting migrant swan!
And the bantam thundered on!
He groped my fan-tail pigeon doves
And my lily-white Columbine
And as I was a lookin' at me budgerigar
He jumped my parrot from behind!
He was sittin' on me shoulder at the time!
But all of a sudden, with a gasp and a gulp
He clapped his wings to his head!
He lay flat on his back with his feet in the air;
My bantam-cock was dead!
And the vultures circled overhead!
What a noble beast!
What a champion cock!
What a way to live and die!
But as I dug him a grave to protect his bones
From those hungry buzzards in the sky
The bantam opened up a sly little eye!
He gave me a wink, and a terrible grin
The way that rapists do....
He said, "Do you see them silly daft buggers up there?
They'll be down in a minnit 'er two!
They'll be down in a minnit 'er two!"
I suspect that bantam Sakata had more influence on go after him than did buzzard Takagawa, and high pincers, therefore, while still common enough in modern play, can't really be said to rule the roost. Although... evolution is like an underground moving river - one of Takagawa's favourite images - and maybe AI is shifting the balance back towards the daft buggers
But to get back to Maître Pathelin's yowes rather than the birds: pincers and tsumes. My conclusion is that they are both useful tools but belong in different toolboxes. They form part of separate nexus. For me, the associations of pincers include sente/gote, tactics, attack, lack of bases, danger of being bitten in the bum, confusion, fashion, joseki books, amateurs. Tsume associations for me are safety, bases, prudence, patience, creating options, strategy, blocking, true control, initiative, pros but not just any pros - top ones and Shusai especially. YMMV.
There! That's filled in a coffee break and a half for you better than work ever does!
If you've still got any energy after that diatribe, or leisure to spare, you might like to listen to the Bantam Cock brilliantly sung. Try https://www.youtube.com/watch?v=B5wXb9XJicM
= hiraki
= tsume
= hirakizume
= hirakizume ?
, approaching the Black stones in the bottom left side at C-08? 
is a mistake that loses 9½% to par, 
stones. 
stone. Then ;b19: continues the attack on the 
in the game is a mistake, losing 9½% to 
= a
makes a small base, and the fight continues. White has managed to complicate the game.