Terminal locations for key functionality, and suggested repair steps

With regards for where to test at, it's hard to say because i'm not aware of any specific testing points given such a small board. What may be a good idea is to get a multimeter & probe some pins on the screen connector. If you do see a voltage, try holding it there & see if the voltage drops out of the screen or if it stays consistent. If it drops out you have a power issue most likely. If however the screen turns blank but the screen fully functions then most likely that SoC is somehow stuffed up & needs a new board .-.

The charging issue would lead to battery but if the watch powers to the home screen ever, and the screen looks like all the pixels work correctly but it just glitches out, then this sounds like a deeper problem.

In summary:

• If screen powers on then off AND it drops in voltage, you have an issue with the power being supplied
• If screen powers on then off AND it does NOT drop in voltage, that is more likely than not an SoC issue
• Battery could be causing either but the test of the screen voltage may be the best way to go
• Also try probing the battery connector to see if the voltage is consistent

Hope this helps let me know how it goes or if you have anything else :)