It's more of a feeling, I know going from a light weight flexy crank to a stronger heavier crank makes my bikes feel like they accelerate quicker, same again going from a Cr-Mo frame to an alloy frame. I went the light weight route on a MTB once and the bike felt horrid so I just bolted the strongest parts I could to it and the bike felt so much more lively even though it was over a kilo heavier.
Yeah most definitely.
You can get similar results by maintaining the gear inches but varying the size of the Chainring/sproket combo - this affects chain tension which in turn flexes the frame more or less.
As you apply pressure to the pedals the frame deflects, storing energy. As the pedal moves down to the 6 O'Clock position, the force you apply weakens and the stored energy is returned the system. This gives unique characteristics to certain bikes - A torsionally stiffer frame will return the energy faster and sooner. This would suit someone with a high cadence for example (maybe - I've yet to see an empirical peer reviewed study - until then its just 'in theory') . So sure choose these things because it feels better.
Just want to add as well that both of the road frames I have made are from the 'least stiff' steel tubing columbus make - small diameter and wall thickness down to .5mm. At no point have I found either bike to behave badly due to the frame not being stiff. In fact both descend like demons. Also note that bolting strong parts onto a frame isn't making the frame stiffer. If anything, stiffer parts will make the frame deflect more. If you do a google search of Bicycle Quarterly they did a very good write up on frame stiffness. It included a double blind test, in which both cyclists agreed that the more flexible bike was the better bike.
I get a nervous twitch every time I see a (road) bicycle manufacturer start rambling on about frame stiffness - its complete BS. In terms of bending due to pedal input, the only people frame stiffness will affect is Track sprinters and BMX racers. For BMX the frame is so small that I can't ever see stiffness being a problem and for the Sprinters its a about confidence not loss of power.
If you look at the modulus of elasticity vs Specific gravity only - you could make a bike (or anything for that matter) out of Steel, Titanium, Carbon Fibre or Spruce, with similar stiffness and weight. The difference would be in both packaging (that would be a lot of spruce compared to Steel) and yield strength and HF vibration qualities.
I believe, the reason for that massive looking structure we are seeing in CF bikes around the BB, are less to do with 'performance' and more to do with reliability.
CF, unlike steel, when it fails does so with not much warning and its nearly always a show stopper. On the other hand Steel will either yield or crack around welds. Even then the crack propagation is slow and tell tales will show up in the form of creaking and rust stains. CF is sensitive to the speed that loads are put through it