I have to force my weight back on a lot of bikes, but for me the issue is more my center of gravity being too far forward relative to the wheels. The solution for me is riding bikes with short chaintays and long front ends. I ride 175 cranks. If I have toe overlap (even if its just the fenders) that almost always means that bike is s so-so compromise for cornering and I have to remember to push my weight back.

The bike I raced was a '76 Fuji Pro. Very short chainstays. No overlap. BB was so high I rarely hit 175 cranks and i pedaled a lot of turns. That bike was scary early season but that was mostly a function of very quick steering. Once I was used to it, I trusted it completely on mountain descents. I never thought to push my weight back. That didn't start happening until I replaced it with a more conventional bike.

Funny this coming up. I see both of these issues now that I am running my Peter Mooney fix gear. I decided to have the bike built with a highish BB. Good move! I love it now after second guessing myself for 35 years. The bike was designed to be able to do everything but race. It has long chainstays for touring loads and big tires. Front end was kept in a bit to keep the wheelbase reasonable. Too short. I have to push back a bunch to corner. Big soft tires, esp in back help a lot. I'm planning to get the 28c Vittoria Graphene tires, hoping they will help.