It's quite possible to have wideband AM radio. Some radio stations did it in the US before the FCC standardized bandwidth and started checking envelopes. Radio Caroline, the UK offshore pirate station (1964-1968), was wideband AM.
Noise on AM can to some extent be overcome with power and a low modulation percentage. That's how analog broadcast TV worked. (Broadcast TV was AM video, FM audio.) The black level for the video signal was well above zero. A high black level allowed showing black areas without excessive noise. About 80% of the RF power went into the carrier because of that. Simple, but inefficient. The same trick can be done with AM audio radio, although it seems that's not done much.
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