Yes, you are definitely missing the holding clip part of the connector. Make sure you don't have a twist in the cable and that pin1 is in the 1 connector position. The way the pins were bent there could have been a short and a on-board fuse could be blown. Hard to say what happened. Gathering all other connectors are properly in place?

Yeah, I checked all the other connectors and they were all firmly in place.

I'm able to get the player to open consistently, and a few times I got the screen to display normal figures. The only problem is that without something to hold it in place, the connector falls out. I was searching for "13 pin ribbon cable connectors" so I could just take the holding clip part off of it and use it on the player, but I can't seem to find any.

If you can solder just take out the connection and solder the cable directly in the board. It is connectorized to allow swapping out boards easily.