Here's a thought: Maybe the local oscillator in the TV's tuner is on the wrong side of the TV channel? TV sound carrier is 4.5MHz higher than the picture carrier, and no sound carrier 4.5MHz lower than the picture carrier.
Try both the VCR modulator output, and the signal generator at the same time. Tune in the VCR video picture. Then sweep the generator (that generates the test sound carrier) at frequencies around that of the TV channels (plus minus say 10MHz). Channel 3's video carrier is at 61.25. The sound carrier is at 65.75MHz. NOT at 56.75MHz, but if you find that you can hear the generator at 56.75MHz for channel 3 and not at 65.75, then I'd start looking at the tuner's local oscillator.
Real TV channels' video signal is AM modulated onto their carriers, but the lower sideband is filtered to stop at 1.25MHz below the picture carrier, but it goes up to 4.2MHz above the picture carreir. TV modulators in VCRs don't bother to filter the lower sideband. So you might have a situation that lets you see good video from the VCR, but fuzzy video from real TV channels, over the air or from cable TV (but not from a set top converter box, but only direct from the cable).
You may also find that the sound IF may be off a little. Using your frequency counter on the generator, zero beat the VCR video carrier with the generator and see what the frequency is. Then note it down, and then sweep the generator to find what frequency it takes to get sound out of the TV. Should be 4.5MHz above the picture carrier. If it's tuned to say 4.7MHz, it will miss the sound signal from the VCR or other source. Tuning the fine tuning knob while ignoring the picture quality might find you the sound signal. Try channel 2 and channel 4 and hunt around with the fine tuning and see if you can find the sound signal. If you find channel 3's sound on channel 4, but no picture, then that could mean that the local oscillator is on the wrong side of the TV channel.
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