When I try to run my LD player through my VHS/DVD combo player (to keep the single SCART input in the back of my TV free and maybe use whatever video built-in video filter it might have), the video looks like this:

https://imgur.com/a/tn1Wz0v

Any idea what's causing this? When I start up the player initially, the blue screen displays correctly (but I have to change the output to PAL so I guess it doesn't like raw NTSC signal either), but when I actually play a disc, it turns into this and toggling between PAL/NTSC just removes the colours.

Looks like some type of sync issue.
Could be the NTSC PAL thing or could be everything syncing with one another.

Any suggestions on how to fix it? I mean, besides just plugging the player straight into the TV.

'fraid that's your best option, the combo player can't deal with PAL60.

To quote Luke Skywalker; "What a piece of junk!"