One argues that the weight of the frame can simply be added to the body’s weight when the rider accelerates or keeps the bicycle in motion. Hence, it doesn’t matter if a specific amount of the weight is subtracted from the body and added to the frame and vice versa.
In general that's correct. If you're accelerating the system (bikle + rider) forward, or you're pushing them up a hill against gravity, then the energy required is proportional to the total mass, plus any losses. Most losses will not be proportional to the mass of the frame, so the distribution doesn't matter much.
a bicycle with a smaller frame weight is significantly easier to e. g. accelerate (especially when steepness increases) compared to one with a heavier weight.
Also true. Now, most riders can't do this experiment where the total mass remains constant. In general, a less massive bike means a less massive system. Unless they're wearing a weighted vest while riding the light bike, I'm don't think the experience contradicts the first one.
Second of all, the rider may be noticing how the lighter bike can easily be accelerated out from underneath them. It might feel "nimble" and light, but still take the same amount of energy to accelerate the entire system. The actual energy expenditure and the rider experience do not have to correlate perfectly.
There are a couple other things you might consider. One would be if there's a different biomechanical response. Your body isn't 100% efficient. Conceivably, there's a reason that the heavier frame would make the body more inefficient. But, I'm not aware of anything that would obviously make this true, and such effects are a bit outside the realm of "physics".
Another thing is the differences in the bikes themselves. It costs more to make a super-light bike. If you're already paying more, you might also pay to have better tires, better bearings, etc. It could be that bike efficiency is correlated with lower mass. So the effects that some would see are real, but not due to the mass difference.
From a physics standpoint: $F=ma$ and $E = Fd$. If the mass is identical in both situations, and the uncorrelated energy losses are the same, then energy expended is the same.