If the FM radio at the receiver mixes with a another frequency to bring down the carrier frequency, wouldn't that affect the message signal?
This is the fundamental thing that all DSP bases on:
Any band-limited signal is 100% accurately representable by a baseband signal of the same bandwidth, and vice versa.
So, no, this doesn't affect the information at all!
In fact, this is the central result of Nyquist's works on the theory of continuous-time signals (with a few additions, it's equivalent to his sampling theorem).
In today's digital world, why the FM radio still uses frequency modulation (FM) and not FSK?
Because of roughly 80 years of legacy devices! FSK is something that has a discrete alphabet of symbols, so it's only applicable to digital transmissions, really. Audio, as is, is an analog signal. So you'd first be converting your analog audio to digital, then modulating it, and then converting it back.
Fact-provenly, the spectral efficiency of digital modes is higher than that of analog FM – but you need digital receivers, which somehow still aren't that widespread.
That's a really long, and partly sad, story of badly introduced standards – there's multiple standards that strive to replace the analog radio broadcasts, and as far as I know, none has so far reached the proliferation that was once envision (note that this applies to terrestrial broadcasts only. For satellite radio, digital modes are overwhelmingly more popular):
- DAB / DAB+: High-quality FM broadcast audio alternative; practically unused by anyone who isn't sitting in a high-end car with the "all-incluse" entertainment system. The difference between being able to receive DAB or not usually is something that the car dealerships earn a couple hundred Euros with. So not attractive at all, although German government has spent a lot of money on pushing that standard. Part of the problem was that early DAB modes had weak error correction, making the reception often worse than what people were used to via FM, receivers are still a lot more expensive, and DAB+, although it solves the quality issue, hasn't been adopted by everyone selling digital receivers. Coverage is mediocre, at best, compared to traditional FM worldwide.
- DRM / DRM+: Digital Radio Mondiale, meant to replace the AM broadcast. Actually stands way better chances at seeing widespread adoption, because it solves actual problems, and the Indian Government alone has a 1MW transmitter to reach rural areas (read: most of India's area), so there's a lot of potential audience. Also possible candidate for broadcasting in Australia. There's about one receiver you can buy in Europe for that (source: been sitting in the same lab as the guys implementing the only FOSS transmitter in Europe).
- add-on standards to "normal" FM: Just as stereo is an add-on to classical mono FM using more bandwidth, there are digital sideband and in-band modem standards (HD Radio etc) that provide additional functionality, like fine-grained weather info etc. Pretty much stillbirths, because where FM SNR is good enough, chances are high that people just get 2G/3G/4G coverage, offering actual advantages over just FM RDS.
By the way, none (to my knowledge) of these standards use FSK – if you're already doing digital radio in the 21st century, you'd typically go for something with higher spectral efficiency (most of these standards are OFDM things with PSK or QAM subcarriers).
So, to conclude, reasons for not adopting digital radio are:
- literally billions of legacy devices that would need to be replaced
- lack of advantage of digital modes ("FM sounds just fine, and I don't need more channels") or need ("99% of the population can get FM radio, and listener numbers are declining, why do we need new technologies")
- actual technical disadvantages (e.g. DAB (not DAB+) was often worse than FM reception, and power consumption is still higher)
- cost of new receivers high
- bad time to market (broadcast standards take much longer to adapt than cellular standards, which now make broadcasting obsolete in metropolitan areas)
- segmentation of market