Simply cannot delete a section break. It is impossible. Cannot be done. Please prove me wrong.

I don't know if this applies in your situation, but one thing I discovered is you cannot remove a section break if you have a cross-reference (Figure, Table, etc.) that crosses the break. Apparently, the break is stored in the cross-reference link and Word is not smart enough to remove it when you delete the break on the page. Delete any cross-references that cross break, then you should be able to delete the break (you may need to turn off Track Changes first as well).