Why does this regenerative receiver not just turn into an oscillator?
It intentionally DOES oscillate.

It goes beyond a regenerative receiver and is called a super-regenerative receiver, invented by Edwin Armstrong in 1922.

The values of R3 and C3 are arranged so that after oscillation starts the voltage builds up on the emitter due to rectification of the oscillations.  After some period (10's of microseconds typically) the bias will be such that the oscillations stop.  The voltage will then decay with the time constant of R3/C3 until oscillations start again.  This typically occurs at a frequency in the range of 10's to hundreds of kilohertz.

Normally in an oscillator the oscillations build up from noise however if any signal at the oscillation frequency is present the oscillations will build up more quickly.  This is the signal being received.  The average supply current will depend upon how quickly the oscillation starts which depends upon the signal present.  Thus the supply current is dependent on the received signal, the output from R1 depends on the supply current.

Usually the transmitter is amplitude modulated at an audio frequency and this can be recovered from C7 with suitable filtering and amplification.

A super-regenerative receiver can get extremely high gain out of a single stage but when no signal is present the output will be very noisy.  Because the build up of oscillations is exponential it usually has a logarithmic response.

They are used less these days because they tend to radiate and also have a very broad bandwidth. A common application is for garage door openers.

kevin