FM modulation, amplitude problem
The probleme is that i'm suppose to have an unitary gain.

You should realize that a properly designed oscillator needs to have a loopgain higher than one (unity) otherwise it can never start-up.

The unitary gain or loopgain = 1 situation will then "happen by itself" as that is the situation where the amplitude remains stable. The oscillator will find this point of operation (maintaining a certain amplitude, loopgain = 1) by itself.

Suppose that the loopgain is always larger than 1, that then means that the amplitude would increase to infinity. Of course, in the real world that cannot happen.

So what does happen then?

Well, the amplitude increases to a point where some stage in the oscillator saturates, for example the common emitter in your circuit reaches its maximum amplitude and if you would apply even more signal to its input, the output would start to clip.

That behavior means that the gain is decreasing! Instead of the, for example, 10 x gain you get for small signals, you get less gain like 5 x for a large signal.

That is the mechanism that will stabilize the amplitude to the point where the loopgain of the oscillator becomes unity.

So you should not be aiming (designing) for a low gain! When I design an oscillator I design it such that the loopgain will always be significantly more than one even, so at least a factor 2. Taking into account variations in supply voltage and temperature I might need to design the loop such that the typical loopgain for small signals equals a factor 5.

Regarding the frequency modulation: you're not using a separate Varicap, instead you're varying the biasing of the NPN transistor which changes the values of its junction capacitances slightly. That works but do not expect that you can vary the frequency a lot. A more standard approach is to make C13 a Varicap so that the LC tank C13, L1, L2 gets a variable resonance frequency.