People can be so petty.

"Kaya", "the Kaya server", and even "kaya gee ess" work perfectly fine in spoken conversation. Kaya is faster to pronounce than "KGS". Kaya is also about as fast to type as KGS. There is no need for an acronym. Other servers need acronyms because they have no sensible names. "Internet Go Server"? How do you shorten that. Same for online-go.com, etc. KGS once stood for "Kiseido Go Server", didn't it? This is still not so helpful because it's "the go server by Kiseido", not "the go server called Kiseido", so you couldn't just call it "Kiseido" either. Now it seems that KGS is a recursive acronym (as the website calls it "The KGS Go Server". Basically, the "acronyms" have become the normal names of these servers.
On the other hand, who uses an acronym for Tygem, or WBaduk? Those servers have sensible, unique, and reasonably short names, so there is no need for an acronym. Same for Kaya. Sure, as the name of a wood it is not _that_ unique, but still rare enough to stand on its own most of the time based on context. When required, it is easy to specify by adding the domain extension.
There is also no chance whatsoever that Google is going to confuse KGS with Kaya.gs. If you expect this, you really don't give Google enough credit for their algorithms. The only way this could even be remotely possible is if KGS would disappear into complete obscurity, but if that were to happen, it wouldn't be Kaya piggy-backing on KGS' popularity, but the other way around.
And even if some newbie would be to make the mistake of calling it KGS once or twice, you can be sure that he would quickly stand corrected, as the real KGS is not likely to disappear from the public consciousness anytime soon.
Kaya.gs is a fine name, one that sounds a lot more harmonious than all of the alternatives that have been suggested so far. Let's not be petty about the initial letter.