I forgot to clarify in my previous post that the "senior citizens exemption" (a/k/a Aged exemption, a/k/a/ "low income senior" exemption) is handled separately from the Enhanced STAR exemption, even if it's not shown that way on some tax bills. So it is
possible that even if a Basic-STAR-only buyer purchases such a house in January or February, the Aged/Low-Income reduction
may still go away before the December tax bill is generated, because changes made in 2019 to how the over-65 types of exemptions are handled might have affected that as well. The Aged/Low-Income exemption is the one that makes the most difference on the tax bill, because it is applied to almost all of the tax line items ... not only the school taxes.
It so happened that I was eligible for the maximum exemption (unlike Enhanced STAR, it's a sliding scale based on one's income during the previous year) several years ago when I was unable to work due to cancer treatment (which is not considered a disability.) That exemption, combined with the normal Enhanced STAR, reduced my tax bill from $9908 "true" to $4025 net. And that was without any other exemptions such as Disabled or Veterans. It was a one-shot tax break but nevertheless a welcome one.