I expected there to be a miracle IC that would just require an audio and carrier wave input
Ah, but FM actively modulates a carrier; you can of course use the external oscillation input as carrier for a superhet design, but then you'll still need to generate the FM-modulated IF and suppress the leakage of the oscillator at the output (wíthout suppressing the frequency-varying intended carrier). That's a rather complex thing to do in a single IC.
I came here to ask why are the radios built always in specific bands like 136-148/200-260/400-430 MHz instead of working continuously - is there a legislative or physical limitation?
Yes :D
both, mostly!
Also: If you only offer specific bands as device manufacturer, you don't have to guarantee performance in between. So, since it's not a big market you'd target with anything that's not commercially legal to do (the couple thousand ham rigs you could sell at most ... pffft).
When you do a superhet FM transmitter, you produce RF at \$f_{\text{LO}} \pm f_{IF}\$ (and of course other harmonics/intermodulation products), where your message signal is actually a frequency-changing \$f_{IF}\$, but you only want the sum, not the difference (or vice versa); to isolate the sum frequency for a clean signal, you will need to filter everything below \$f_{LO}+f_{IF}\$. That only works with a fixed filter bank if you can't pick from more than an octave of \$f_{LO}\$.
And my second question is whether is there a way to approach this problem that would be friendly for someone who usually works with digital stuff (like an IC or module) instead of analog/radio electronics.
Sure; you could generate an IF signal with e.g. a microcontroller (FM modulation of a carrier between say 100 Hz and 75.1 kHz is not that mathematically hard to do); then, mix that up with about any LO (you can buy digitally controllable oscillators, Silabs has such) using about any mixer (SA612 is certainly a classic). Then, you get all the intermodulation products, and your filtering needs to select the one you actually want to transmit.
A very digital way to do that is the rpi_tx software, which uses the PWM units on a raspberry Pi SoC as generator for an RF signal; you'll have to add a solid amount of filtering to get rid of these harmonics you don't want (you only want exactly one of them).