The fact that in Drake's main era, self-destruction seemed much more likely should maybe give one pause to wonder if similar levels of confidence in a Singularity should be justified.

I agree.

For example, while Drake just needs the likelyhood of a star to have planets, you would need information about the distribution of such stars. Star systems near the galactic core in many ways look different than those near here.

For our purposes, we'd only have to look at the distribution around us, maybe something like all stars less than 1000 light years away that aren't brown dwarfs. We know those pretty well.