I don't think the centre frequency is going to be very stable anyway,as the thing is self referencing.
I'm at a loss to see how the tuning register can stabilise this,as it must ultimately be dependent on the RC oscillator stability.
If we in turn mess with it to produce FM,any stabilising effect will be compromised.
To get a sine wave output,you would need a BPF,the bandwidth depending on how much you want to shift the frequency.
This kind of reminds me of something we discovered when we were using an "El Cheapo " video modulator kit back in the 70s.
(We put a crosshatch signal in to produce a modulated signal to use in setting up convergence on a TV set.)
Even without power on,you could still see the crosshatch on the TV,as there were enough harmonics generated from the video to appear as RF.
We thought we had found a neat trick,& played about with it for a while,but it wasn't really useful,as the TV didn't lock very well,& it was quite noisy.
In the same way,the "FM transmitter" idea is interesting,but impractical.
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