Thanks for the many ideas!
I already built the elenco kit. Very nice, but strange combination - on the one hand, they explain how to read resistor color rings, on the other hand I have serious doubts that without understanding of how a FM superhet works and a decent set of instruments (RF source with modulation, scope or spectrum analyser) you get a single tone out of the FM part. Far from beginner level. With proper instrumentation adjustment is more or less straightforward, although pointless in the AM case - after I had finished the kit, I found out there are no more AM transmitters left. Shortwave radio is a different story though, but this kit does not cover SW.
My initial idea for designing a FM radio was to practise impedance matching, VNA work and maybe some smith chart magic. As it seems all discrete FM tuners use these square "filter cans" in various colors, which cover both inter-stage impedance matching and usually also filtering. These completely vanished from the face of the earth - including data sheets or design data. So one could design those filter/transformers from scratch - which I feel is effort going out of hand.
One interesting problem remaining would be RDS (RBDS) decoding. Have a look at the spectrum after the FM demodulator (from chinese kit):
Below the 19 kHz pilot tone there is the L+R signal, then left and right of the suppressed 38 kHz carrier the L-R signal for stereo (here you can see the signal is actually stereo). Above (second cursor) at 57 kHz there's the RDS data stream, 1187.5 Hz 2-PSK modulated.
https://en.wikipedia.org/wiki/Radio_Data_SystemI wonder how one could demodulate that (not using a RDS decoder chip like TDA7330). Maybe a phase locked 38 kHz pilot tone is already available from the stereo decoder, if that would help.
SI4706: Amazing - just connect antenna and SPI interface - but where's the fun in that?
Maybe have a little fun and use a higher first IF, or upconvert first.
Interesting idea. Could you explain how you would do that?
TinySA also does an upconversion
https://tinysa.org/wiki/pmwiki.php?n=Main.TechnicalDescriptionto 433 MHz, then to second IF 870 kHz. I guess those values come from what filters are available.