How to isolate overlapping FM signals?
What if they did all use the same carrier?


What you're describing is a single-frequency network (SFN). These are in common use for things that are not stupid FM broadcast.

The whole truth is that reception from different senders in an SFN just look like heavy multipath propagation, where the sent signal just takes multiple paths of different length to the receiver, and thus the receiver sees the signal as what would come out of a tapped delay-line.

If you consider it like that, the effect is clear: the different delays with which the signal can reach the receiver lead to something that for all practical purposes is an FIR filter. If you look at it from the viewpoint of digital communications, you get Inter-Symbol Interference (ISI) as delayed versions the last sent symbol still reach the receiver while the shorter paths already brought on the next symbol.

If you look at the same thing from the perspective of spectrum analysis, it is a frequency-selective channel. 

Either way you look at it, you counter that with an equalizer that requires knowledge of how the channel looks (i.e. the different propagation delays, or the frequency-dependent behaviour of your channel, which are just two descriptions of the same thing, linked by the Fourier transform) and can then do its best to reverse that effect.

Now, sadly, FM radio is not digital communications. You don't get symbols, and by the very nature of FM  being a modulated carrier, you don't even get any kind of "known, not very repetitive" input with which you can estimate what your channel looks like.

This needn't be a problem – if the resulting delays don't make the channel too harsh, you don't lose much SNR at the receiver. We can measure that by comparing the delay spread (difference between shortest and longest path) to the inverse of the channel bandwidth. If the channel bandwidth is "sufficiently" smaller than the inverse delay spread, we're probably in luck and can still consider the channel "sufficiently" flat. "Sufficiency" here depending on your needs, so let's just do a order-of-magnitudes calculation.

An FM channel is roundabout 150 kHz wide. Invert that, and you get $6\frac23$ µs. 

Electromagnetic waves travel with the speed of light in vacuum, and for all practical purposes here, in air, too. So those microseconds correspond to a distance difference to two transmitters of $c_0\cdot \frac{20}3\cdot10^{-6}\,\text{s}=3\cdot 10^{8}\frac{\text{m}}{\text{s}}\cdot \frac{2}3\cdot10^{-5}\,\text{s}=2\,\text{km}$.

Since we can be pretty sure that different FM radio transmitters aren't only 2 km apart, we can directly see that the resulting radio channel will be wildly non-flat and frequency selective. In fact, since the distance between these will be orders of magnitudes larger, we can assume that even given good channel information, we might have frequency-dependent deep fades that we simply can't properly "equalize away" (you get nothing through at those frequencies, so you can't even amplify anything to make it "flat" again).

In other words, FM and SFN don't play well together. There's been attempts to still implement SFNs for FM broadcast (I think Netherlands tried that), but as far as I know, they failed – they can only work by "aligning" the interference for limited locations, and if your network can't be too large and there's no strong variation of propagation environments for different transmitters (which would explain why small and flat Netherlands was an attempted adopter).

That's one of the most important reasons the world is leaving analog broadcasting behind. DAB and DAB+ are SFN-based, as their heavily multi-carrier oriented OFDM system with channel bandwidths of 1 kHz allows for much larger delay spread, and their frame structure and channel coding allow for heavy error correction, so that even deeply faded subcarriers don't matter that much.

So, yes, you've found a (if not the most important) spectrum management reason why FM should, if you ask me, die as soon as possible: it enforces the non-reuse of spectrum, and that's a waste of a precious resource.

Now, you mention multiple antennas: 

Well, yes, if you have multiple antennas, you can use phasing to make them have some directivity, i.e. you can "simulate" having a single antenna that only receives from one direction. Hm. You buy multiple antennas, build a complex and large phased array and try to operate in a SFN, and the best-case scenario is you get the same performance as in a MFN with a single antenna. Doesn't sound too tempting, and, you'd need to have an idea in which direction you want to "look". It's notoriously hard to estimate that out of, again, an analogue signal with no known non-periodic structure. In a situation where you're at a fixed position (moving obviously breaks knowledge of directions) and adjust antenna phases just to get broadcast radio: Have you considered satellite radio?