Well, yes and no. It's a bit more complex than just taking the numbers from Shannon-Hartley, but I admit that my original description was at best lacking, so thank you for pointing that out.
Shannon-Hartley describes that the theoretical information capacity of a signal given a bandwidth and an SNR. Doubling bandwidth halves your SNR (received noise increases, received signal does not), in turn reducing the bits gained per unit of bandwidth. At very high SNR, doubling bandwidth almost doubles capacity, but as SNR goes down, the benefit of additional bandwidth levels off until bandwidth no longer has any effect.
However, this provides the number achievable by a perfect modulation scheme using all available bandwidth and signal strength. AM and FM are both incredibly inefficient, and more importantly have very different reactions to noise - something Shannon-Hartley does not concern itself with.
With truly random noise, FM and AM noise both scale based on noise amplitude as you say. In AM, all noise overlapping with the band is played back verbatim, whereas in FM only the noise causing frequency variations in the carrier wave have any effect on the signal, and end up with a non-linear response to noise.
However, we do not deal with purely white noise, and FM has far superior handling of non-random noise. In order to have any effect, it need to either induce frequency shifts to the carrier wave, or have enough power to cause the interference to be captured instead. There's also the far higher power efficiency, as FM puts all its power into the signal, whereas traditional AM puts most of it into a useless carrier and wastes half the remaining power on the redundant sideband (yes, SSB is a thing). These were certainly also factors in FMs demise.
A simpler means to remove bandwidth from the equation would be to compare with a narrow-band FM transmission, or by multiplying the input waveform for an AM transmitter by some factor to fill the bandwidth. I believe FM should still handily beat it at least above its threshold. I don't see anyone giving exact numbers of this though, so I guess it could be a fun SDR project for someone wanting to prove either point. :)
(Neither AM nor FM is of anything but historic value at this point - their only redeeming quality is discrete circuit simplicity if you need to MacGyver one out of shoelace and bubblegum, but that's it.)