In terms of Tsumego, there's no doubt that it CAN improve reading. The debate seems to be clicking before you have the variation worked out, versus clicking through the variation first and then examining the result.
Tsumego is a tool, though. I'm going to try and describe my ideas here using a metaphorical structure, though the metaphor is overly simplistic, I think. Hopefully, it makes sense.
Let's say it's a screwdriver, for simplicity's sake. Now, let's say Player Alpha's learning process for reading can be represented an infinite set of screws, and it improves by tightening each screw. The process of tightening represents the visualization before playing out the stone. Once the screw is "tight", we can say that Player Alpha is "reasonably sure" of the solution and can play it out, satisfied that he/she got the correct answer. The tightened screw represents a tight hold on a particular concept. It takes a bit of effort, but the screw will hold and won't loosen easily, especially if you tighten more screws close by, sharing the task of holding related grey-matter ideas in place.
Let us now consider Player Beta, who's learning process for reading can be represented by an infinite set of closed paint cans. "Clicking Through" a tsumego is the act of opening one can with our tsumego brand screwdriver. Unlike the screws, all this does is expose the ideas and allow them to be examined. There is less initial investment in effort involved (though some cans may be harder to open than others). The next step for this player is to make sure the idea paint inside doesn't evaporate. Beta needs to paint (play games using the idea) and needs to think about what designs he can do with the paint as well (thinking about variations AFTER looking at the solution).
Clicking through alone, as Kirby said only wastes the paint. However, some people just learn better by opening up the idea first, and then building on and improving their internal structure. This requires a greater flexibility with memory (and possibly capacity as well). Other people learn better by stretching their internal structure to map to unfamiliar situations, finding similarities and building possibilities first. This requires greater analytical skill.
Both Alpha and Beta improve their visualization ... Alpha builds an internal mind-structure which represents his reading skill; Beta paints an internal mind-picture that represents his reading skill.
I don't real people are as polarized as this, though. I think that there is benefit for players wishing to improve to use both methods, depending on the results they get from applying either strategy.
End message: don't be lazy. If you "click through" a tsumego solution, you are not done with it. Arguably, for some sets of tsumego, even if you have a "solution" thoguht out before you place your first stone (along with any number of variations, you're still not done with it unless you understand what you got out of solving it.
Post scriptum: I am fairly good at tsumego, but I learn more like Beta.
