(If anybody complains that the expectation of some Singularity-like development is ideological: no, it is a reasonable guess based on the current evidence, much like Drake's expectation of every technological civilization's eventual self-destruction was reasonable in his Cold War era.)

The issue isn't whether it is "ideological" but whether it is likely. And how likely matters a lot in this sort of context. 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.

It is also worth noting that to get the sort of average you want we need probably more precise data than just that in the Drake equation. Even that equation is only an approximation (among other issues it assumes that all the variables are independent). What you want to do would likely hinge even more on sensitivity to those sorts of issues. (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.)