And, sadly, most of the interference issues would not be solved by installing a filter on the antenna,
That wouldn't help at all because the issue is the TV being desensitized by a signal that is out-of-band for TV, but in-band for Ham radio. You cannot filter out the signal that you're trying to transmit, as that defeats the purpose.
If the signal that is interfering with the TV is coming in the antenna connection, then installing a filter on that connection is exactly how you'd solve the problem.
Again -- absolutely not. I'm going to try to clarify this for you one last time.
This is a case where the Ham is transmitting on a ham band (not on a TV band), but the proximity between the Ham and the neighbor's TV antenna means that the neighbor's TV, which lacks an input filter to initially reject the Ham band, is desensitized by the powerful signal, and as such isn't able to receive the weaker broacast TV signal. The effect the neighbor sees is "I can't see my channel", but it's neither the Ham's nor the neighbor's fault as to why that is. It's also perfectly legal for the Ham to be transmitting, because the Ham is not transmitting outside of the Ham band, and is thus not interfering.
This cannot be filtered at the Ham antenna. The filtering has to be done at the neighbor's TV.
Yes, you'd filter out the signal that the ham is transmitting whether it is direct interference (the most common) or desense.