I've ridden heavy bikes, and I've ridden light bikes. Light bikes feel better, rotational weight mattering more than anything else.
A bike's does not require gyroscopic effects to stay up -- there was an experiment with counter rotating wheels (to cancel the gyroscopic moment) that was ridable. The actual stability depends on the geometric and pneumatic trail, flop, weight distribution on the steering axis, and a few other things.
There is a middle ground that people have converged to -- Somewhere in the range of around 600mm bead seat diameter and an inch or two of tire. (26"->700c, 25->50mm width covers most of the biking world.)
Also gyroscopic effect has very little to do with the bike staying upright. The reason a bike stays upright is because of the trail interaction and camber thrust. When the bike leans left, camber thrust of the tire makes it turn left, which moves bottom of the bike to the left, correcting the initial lean, while the geometric trail prevents the tire from turning left completely.
When the bike gets too light you feel the effect of bumps and terrain much more and it throws the bike around more (especially if you are "getting light" over said bumps). The lightest bikes also tend to be made for racing and can sometimes be on the ultra super stiff end of the spectrum.
I have two relatives who have raced bikes in the past at high levels, and both owned stores. One of them was a stronger racer who didn't care about gear as much. The other was still a heavy hitter, but wasn't as naturally talented, and enjoyed the nuances between components way more. He and I seem to agree on things.
I do know that tire pressure and width make a huge difference in plushness, but I still prefer a springier ride-feel from the steel I like, vs. even a modern steel gravel bike with 42's with lower PSI. I had such high hopes for that ride, but man, it just felt kinda muted and leaden, and I missed my steel road bike.
But yeah everyone will have their personal prefs, subjective interpretations, and theories about what works for them and doesn't. Gimme that springy lightweight steel any day.
I could see all the mini accelerations adding up to a much larger energy expenditure than would be indicated by a spherical cow in a vacuum.
If the mass of the wheel is 100% located at the rim, the energy goes 1/2 into straight line energy and half into rotational. But -- if you slow down, you don't lose that energy, it just redistributes to gravitational potential. The flywheel effect from heavy wheels may actually help, as it tends to smooth out pedal strokes.
The energy to go up a hill is just mgh - friction. Some cadences are less efficient than others (e.g. for me, standing is more power but lower efficiency), and bikes that are too stiff may not help you get the best rhythm.
In a totally unscientific test -- I've got a 12kg Aluminum gravel bike, wide tires, rack, fenders. I've also got an 8kg carbon racing bike, skinny tires, aero rims, etc. They have similar riding positions (reach/drop), though the gravel bike has a wider seat, which is better for one climbing cadence. I usually ride the gravel bike these days, because comfort.
Two weekends in a row, I did a 12x rep climbing workout (2.5 hours), one on each bike. First weekend, Road bike. It felt _fast_. Quick, lively, accelerated from the stoplights on the way to the hill. Second week, back on the gravel bike, grinding up the hill. Total ET difference: 5 seconds.
Sure there are differences.
