Um the L in LAN stands for “Local”. Sorry I’m from the department of redundancy department just conducting an inspection here.

True, but actually I wanted to mean that if you're out of the 1st router it might be too slow already (especially for something like GIMP)

Fwiw, I was trying to do this over LAN. In fact to a headless machine in the same office. This kind of basic thing should just work.

I use VNC all the time, but sometimes it is just more convenient to use a remote X connection.

It was 20 years ago, I tried running GIMP over remote X on a 100base-T LAN. It was pretty much un-useable. Granted modern day Gigabit networks might have made it more palatable, but I suspect with all the modern UI toolkit assumptions about the X server being on the same machine, the end result is that it will still be worse than an optimized VNC.

As a comparison, windows remote desktop on the same network was pretty snappy at the time.

Very true, but there is no excuse for prioritizing performance over correctness.