I live in St. Louis and you can still get decent houses in nice neighborhoods in the 200 - 300k range though it is getting a bit harder.

A few things I have observed over the years you might want to check. Take a flashlight or something and look at the actual contacts in the port and see if they are bent. Sometimes the old cheap 4 in one carts have poorly made PCBs what arent beveled correctly or are too thick and squish the pins and they no longer make good contact. It looks like an official one in the pic but unless you have had this since new who knows what someone threw in here. see here for some more info on that

Also these backup carts do just sometimes go bad though I don't know if I have ever seen an official one go bad but buying a new one to test may help with that as they aren't that expensive. They also usually have lots of built in features like an action replay and acting as an extended ram cart. Just make sure to try and check that the pcb is made correctly with that link above

Depending on your hardware you should be able to have Handbrake use your GPU to reencode the video much faster than your CPU. If you have Nvidia it would be Nvenc, Intel is Quicksync, and AMD is VCE. If you select one of those as your codec it should go much faster. Check the hardware encoders section on the Handbrake documents https://handbrake.fr/docs/en/1.4.0/technical/video-nvenc.html . Even if you were using windows you would run into the same problem at some point you are limited by how fast your hardware can process the video and no software can make up for that.