Identifying an old RF circuit
It looks like an RF transmitter to me, maybe FM
It smells of amplitude modulation rather than FM because of the way 2 modulates the base bias in the oscillator.
Moreover, I'm puzzled by the blocks 3a and 3b. Are they input(s)? Why are there two nearly (different resistors values) symmetrical blocks?
3A and 3B are two independent phase shift oscillators that amplitude modulate at two different frequencies simultaneously thus giving the transmission some degree of integrity. The receiver will likely demodulate the transmission and only operate (maybe a garage door or lock) when it is getting both baseband frequencies.
Is the block n°1 a known topology? If so, what's its name? Is the crystal oscillator fixing the carrier frequency at 27.120 MHz?
The oscillator is a type of hartley oscillator that uses a crystal to lock-in the frequency of operation.
Where does the antenna should be connected ?
It quite possibly uses the split coil to generate just a localized magnetic field rather than a full EM field. This works at much less distance than an EM radiation because only one field is being transmitted. If you don't have an antenna connection then that settles it - garage door opener is the likely product.
Finally, if I'm right on the power rails, is there a safe voltage I could apply to test the circuit? The 20 µF capacitor is rated for 10 V.
It's likely it will work at round about 6 volts. You could try using a bench supply to raise the voltage from 0 volts to about 8 volts but don't go higher given the capacitor is rated only at 10 volts.
I can't manage to make the two phase-shift oscillators oscillate.
Quite likely is that the 22 uF capacitors on each have gone very leaky and are taking enough current through their respective pull-up resistors to render each phase shift oscillator "silent". Try measuring the voltage across them and report back.
It's Citizen's Band (about 27MHz) AM transmitter which sends a dual frequency tone - maybe it's intended to activate or turn off some machine.
The crystal stabilizes the frequency, it cannot be FM. Two phase shift oscillators generate the modulating signal. It's possible to adjust the frequencies of the modulating signals.
The modulation isn't especially effective. It reduces substantially the output RF power, but it can still be ok for the intended purpose. At least it's simple.
Better modulation principle is to swing the operating DC voltage of the RF oscillator by adding an AC component to it with a transformer.
The antenna would be connected through few pF capacitor to the collector of the leftmost transistor. Because it's an oscillator, not much power can be sucked out of the circuit. Otherwise the oscillator stops. The antenna definitely cannot have max. power transmission matching, it can be connected only very loosely to the circuit.
After seeing the photo: A circuit this big with no shielding very likely is easy to detect with a normal CB receiver at 10 meters. An antenna probably has never been used.