The ceramic filter isn't a good solution, they typically have a bandwidth of 250KHz which is fine to filter an existing wideband FM signal but when you put them in a feedback loop it will oscillate at any dominant frequency within that bandwidth. You could be +/- 125Khz of center frequency and therefore on the edge of the sound filter response inside the TV.
I agree with Bob60's value assumptions, I think they should be much higher in value and suspect the 10pF capacitors should be 10nF but that doesn't solve all it's problems.
Starting from scratch, but following the same basic design, I would use the 741 (horrible IC!) in a fixed gain configuration and wire the level control like a conventional volume control, that lets you ground one side of it and also gives a more predictable input impedance. In the oscillator stage I would make the variable capacitor 10pF and add a 10pF fixed capacitor across it, this will allow more precise tuning, the inductor I would drop from 10mH to 5nH (2 million times smaller!). The audio modulates the frequency by changing the base current and predominantly B-E capacitance so you still get FM sound. In a normal TV, pre-emphasis is applied to the sound before the modulator so you may find it sounds muffled. You can add it with an RC network around the op-amp or at the input if necessary.
The video is more of a problem, you want it to be AM and not change the carrier frequency. The present design modulates in a mix of AM and FM with the FM coming from junction capacitances in the same way as the sound works. There is no simple fix but the best without more complex circuitry would be to add an additional transistor after the oscillator and apply the modulation there. In a better design you would add a buffer stage between them so the capacitance changes in the modulator stage are not coupled back to the oscillator. One transistor is better than none though, so replace the video level potentiometer with a fixed resistor and change the antenna coupling capacitor to one much smaller to reduce the effect a load can have on frequency, I suggest 2.2pF. The modulator/output stage would be a conventional amplifier with say 1K emitter resistor in parallel with 100pF to ground the RF components. The video input would go to the top of a 100 Ohm potentiometer with a 220Ohm resistor in parallel with it, this will give a better match to the video feeding it. The wiper of the potentiometer can go to the emiter resistor through a 100uF capacitor. The input comes from the old antenna coupling capacitor. Take the new output to the antenna from the collector through a 10pF capacitor.
As with sound, it is normal to use video pre-emphasis at the transmitter. Basically, in both cases it boosts the higher frequencies in the sound and video before transmission and in the TV receiver the reverse is done. The overall effect is a flat response but noise picked up in the transmission path is reduced by the receiver filtering. With AM video you may not need pre-emphasis unless your picture appears blurred but the FM sound might lack high notes unless you use it.
Brian.
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We cross posted, if you use the schematic in post #13 you can omit the transformer secondary and connect the capacitor directly to the collector of T2. It will not work as well but it avoids a transformer. Please bear in mind that you need the inductor to tune the oscillator to 5.5MHz and the secondary only has to be maybe 2 turns or wire around it.
Brian.