What you could try to do is solder wires to points of interest on the board, such as the regulator and CPU power pins and run them outside the enclosure so you can take measurements while it's running. Just make sure you label everything and don't short anything out obviously. Overheating parts usually make a distinctive smell, that's why I asked about that.

I'm mostly going by what I remember being issues with a regular NES, and I'm pretty sure most freezing issues are either due to bad connections or power problems. It certainly wouldn't hurt to clean every connector again just to be sure. If it was a chip going bad, swapping one out from a working NES would be a pain but doable. You would just have to be careful not to pull up any pads or traces.

Are you talking about the RF modulator on the original? I think the power jack is inside of there. Yours shouldn't need one, at least I certainly hope not. Are they running composite to the TV? Like I said, I'm not too familiar with these...other than it's a cool collectors piece that would be a shame to have not work anymore.