but I'm not sure USB-C output exists.
Not that I'm aware of, no. It's either HDMI or (if you getting the older Blu-ray player) one has Composite or Component output.
Now I'm also questioning the need of BluRay/image upgrade, as it shouldn't be too expensive.
You can always get it second-hand which would be cheaper than getting it new and make sure you don't connect to the internet.
Also as someone that has BluRay player, I prefer BluRay over DVD disc for it being in HD and also, it can still play DVDs. If you already have BluRay collection or don't mind spending a bit to get it on that format, go for it.