One outstanding design question is "how much gain does the receiver need?" In order to figure that out, it will be useful to know how much signal a broadcast FM signal might generate at the antenna. I figured I could estimate that by sticking an "antenna" (random wire) on my scope, setting the input to 50 ohms, and looking at the FFT of the broadcast FM band. So I did that, and here's what I saw:

The peaks here correspond to stations that the FM radio on my phone finds, so I'm confident I'm looking at signal.

The strongest signals have a magnitude of around -70 dBV RMS and weaker signals more like -90 dBV RMS.

I'll need to see something like 0 dBV RMS (about 2.8V peak to peak) at the ADC, meaning I'll need around 70-90 dB of gain.

While this is a crude estimate, it at least gives me a general sense of what I need to aim for, which is good enough for now.