The original code initially changed all date pickers to rich text controls, and then attempted to change them back to date pickers once the date had been changed to an ordinal.
".Type = wdContentControlDate" needed to be moved into a later IF statement to actually do so. I also moved ".Type = wdContentControlRichText" so it would only change the relevant date pickers, but that was really just because, more than out of necessity.

To be fair, I can't see why it actually didn't work for me without the changes, I'm just happy now it does. If it makes any difference, the date I was transforming to an ordinal was formatted simply as "d", rather than the day-month-year format posed in the original question. I ran it in Office 2016, if that makes any difference.