Should I attempt to fix a faulty USB-C port?

I've replaced plenty of USB ports, including USB-A, micro and mini, but USB-C ports are a whole different animal. A small percentage may have all pins accessible with a soldering iron, but I've seen that the majority of them can only be soldered with a hot air station, as there are pins underneath that can't be reached with an iron. Whether it's surface mount or through hole will also factor into the difficulty; you might be able to unsolder all of the pins if it's a through hole mount, but those are becoming rare on modern components.

If you can post a picture of the port as it's mounted on the circuit board, both sides, we may be able to make a better guess, but at a minimum I'd say you'll probably need a heat gun with a fine tip and some Kapton heat-resistant tape to mask off surrounding components from the heat gun. Flux and solder wick will come in handy too. If you have a local shop that'll do it for a reasonable price, I'd personally consider that.

The connector is a thru hole. However, unsoldering this from a multilayer bd is not going to be trivial.

I might be easier with a heat gun. But you still need to clean out the holes. the bigger frame holes will be relatively easy. Of course, that assumes you can buy the part.

However, the whole board is available on eBay for $27.95