Here are typical screen shots of subsequent pages.
I thought I had chanced on something, changing the table caption on the Frame setting from text wrapping to None, which worked on one of my tables, but obviously not on this one.
You'll see the table heading here doesn't repeat on the next page., either Under Table Properties, I have the repeat headings set up just as you instruct. Working as a contractor for NOAA, as with most of the federal agencies where I have worked, we don't have the latest version of Word and can only change headings to repeat under Table Properties. I never use manual page breaks, except between document sections, so that is not the issue, I do have Keep with Next checked under Paragraph, Line and Page Breaks.
Stefan says having Keep with Next checked is what is moving the tall tables to the next page. I've tried unclicking Keep with Next, which changed nothing about the table's position. and tried putting the caption beneath the table. I've also tried creating a table within the same frame as the caption, and that didn't work either.

