Should a graduating doctoral student "care" about his or her advisor's tenure considerations?
I think your approach to this situation is a bit misguided. Yes, you should absolutely care about your advisor’s success (and he should care about yours). But “caring” doesn’t mean that you should sacrifice your own career and self-interest to the cause of helping your advisor, just as, by the same token, I wouldn’t expect your advisor to sacrifice himself for you either. Thus, the way you frame the question as a dichotomy between the extremes of “couldn’t care less” and “will do anything to help my advisor get tenure” is a false dichotomy.

The truth lies somewhere in between those extremes, but broadly speaking I tend to agree with your assessment that the advisor is unreasonable to have an expectation that you will defer the launch of your career for a semester and give up money, health insurance and professional opportunities to stick around and help him finish a project that is no longer a priority for you, for whatever reason. You need to stand up for yourself and explain what your priorities are. But do it in a civil and polite way that shows that you are grateful for the mentoring and other help he has given you (and may yet give in the future through letters of reference etc). Don’t say you don’t care (and better yet, don’t even think it). Do say that you would be willing to make reasonable efforts to ensure a smooth handoff of the project, as you seem to have been doing already. And finally, as @aeismaeil said in his answer, don’t mention the tenure or the grant, since they are simply irrelevant to the discussion. Good luck!