For use in a product detector, is there an obvious "better choice"
between the 6BE6 and the 6BY6?

The 6BE6 seems more common and looks like it'd be better for use with
AGC applied, but that's not how my product detector works, so I don't
know if there's something else which is a determining factor.

The 6BY6 seems to have originally been intended as a sync separator
for TV use.

Tim.