This is the picture frame video? If so I think the final thickness in the video is near 3/4. Could start from pre-milled 3/4 (aka 1″) or rough 7/8 (aka 1″ or 4quarter).
I think it is going to be a bit harder to make a frame less than 3/4 final thickness. Things get tight and thin. Also have to consider how much depth you need for mounting.
Personally I have found 3/4 to be barely enough for fitting a protective glass (actually I use plastic for shipping/kids throwing stuff/earthquake reasons) front + masking + the art + 1/4 inch backing + the little pins to hold it all into the frame. I guess it wouldn’t be too hard to find a backing that was 1/8 for a smaller frame, however.
I don’t think you will regret having made the jig for 3/4 even if you end up preferring thicker and making another jig simply because a frame is a good way to use some left over 3/4 material.