I agree that printing all 5000 is ludicrous, however if you are unable to convince Mrs F then I guess you need to decide what it is you really need. If you just want 6 x 4" prints then a bulk order via the cheapest consumer online printer such as Snapfish, Aldi or Photobox. if you want professional prints then I have always used
https://www.loxleycolour.com/
However, my suggestion would be to first sort out your backup solution, as in immediately right now, this second. I would suggest a physical backup to another drive PLUS an online solution such as OneDrive, Google Drive or even if you are an Amazon Prime customer their free service. You need to do this even if you do decide to print all the photos. Next, I would create a Photobook using some carefully created images as an example as to what can be done. A photobook in my view is a million times better than a stack of snaps, plus it can actually work out cheaper.
I create a yearly photobook for the family every January documenting the year, they certainly get used a lot more than any pile of snaps. I use mixbook.com