After a brief chat with a guy who games in different circles than I do, I think I understand why I'm having a problem with this - I come from a Warmachine background, so, when I read rules, I want to follow them step by step in strict order. I described the responses given to my original post, saying, "it seems like they see the paying of the health token as firmly linked to getting access to the ability - as if the text of the card comes off the card and hovers in a que while the incap procedure resolves, then that text comes back into play, whether or not the card it's written on is still in play." My buddy looked at me for a second, then pointed out that I just described the 'stack' concept from Magic the Gathering. Looking at the incap procedure as a series of effects that are placed on the stack, then resolved from most recent to first does allow Tachyon to pay her last health and then use Push the Limits. Looking at the incap procedure as something that interrupts play when a character goes to 0 health results in her not having any active power cards, despite having paid a cost trying to activate one, and therefore no Pushing the Limits. So, I get where the other opinion is coming from now... however...
This discussion has made me seriously read the rules & FAQ regarding incaps and paying health costs - there are several examples / rulings that show that paying the cost for an ability does not equal gaining use of that ability (usually due to incap). I still read that first page of the FAQ (with it's link to the forum post) as being clear that paying the cost of an ability does not grant use of that ability past being incaped. Here's the text, pasted directly from the FAQ:
"if a character with only one health point left activates an ability which costs a health point, they would pay the cost first, which would result in them having zero health points and therefore being incapacitated before the ability’s effects ever occurred. they would not activate the ability, have it do its effect, and then become incapacitated afterwards." (bold selection is my formatting)
To say that Tachyon gets around this by both incapping & getting the effect at the same time seems to not fit with that description.
The second item I'll bring up is that several times people have posted that 'incapped characters cannot make attacks'... and, while I agree that it is certainly the intent of the desingers for an incapped character to actually BE incapacitated per the common meaning of the word, there's nothing in either the main rules or the FAQ that actually says incapped characters can't make attacks. There are rules about in game effects like hazards leave play when the power card isn't active anymore, and it's rare that a character would be incapped during their own turn (but certainly possible due to redirects, hazards, & more). However, if we go with the 'stack' concept, say that AbZero pays for his thermal shockwave (thus, putting it on the stack, independent of the power card) I don't see why he wouldn't make the attack even if he's otherwise 'dead.'
I'll note, I'm not advocating for 'zombie attacks' - but I do think that using a character's last health and gaining a power use at the same time has more problems than just having the incap procedure work as a hard interrupt of that character's turn. I also see it limiting design space for future health abilities that shouldn't be used after being incapped. I'm okay playing it either way, as long as I understand it beforehand.