On a couple of my bikes I can see the chain move around a bit inside the FD when I start cranking on it. Is this bottom bracket flex, or flex elsewhere? Everything seems tight (grab it by hand, no movement; no clunking either) but in certain gear combos I can get derailler rubbing. It's enough that I'm considering switching to friction front for this one MTB I have, as it doesn't have a trim feature like my Tiagra brifter does (on my road bike that is).

I've been tempted to mess around with my old Panasonic road bike, make a utility bike to tow a trailer; but I recall it being quite flexy down there--I can't imagine it being any better with stronger legs and more loading.

Could be BB or wonky chainrings. Does it happen only when you're mashing? Then it's probably BB.

Are the chainrings true? Drop the chain off the inside ring onto the BB and spin the crank by itself. Look down and use the FD as a guide. How straight does the chainring spin?

To test BB bearing-preload, line up one of the cranks next to a chainstay. Grab the chainstay with one hand and push the crank at the pedal end in and out to feel for play. You can also use the seat-tube as well, but its distance away from the crank makes it hard to see movement.

Chain rubbing the FD as you sweep across the rear-cluster is common. To have the largest sweep-range possible without trimming, you'll want to set your FD height and rotation precisely right as shown on this Park Tool - Front Derailleur adjustment guide.

Trust me, there's no way you're strong enough to flex that bb spindle. Chainring lateral movement is normal.

Yeah, but what about the frame's bottom bracket shell?

Bicycling Magazine used to have a fixture, that was designed by Gary Klein, to measure bottom bracket shell flex.

Yeah, but enough to cause visible amounts of lateral movement relative to the FD cage?