Those connections are called IDC. Insulation displacement connector. All of mine pull right off the boards. The header pins should definitely be soldered to the board.

One of these is a handy tool.

https://www.pinballlife.com/0156-396mm-idc-wire-insertion-tool.html

I believe if you switch to the molex then you will need the terminal ends to crimp on the wire to insert it into the connector. Most people would switch to molex because it’s probably more reliable. But that’s up to you.

The idc push tool would not work for the crimp style molex.

Usually when changing the connectors people will switch the header pins on the board as well.

I would try hitting the board pins with sand paper before any of that

Oxidation/goo/filth can occur on the pins on the board. Many problems are 'fixed' easily enough by taking sandpaper (400 is well) and scuffing up all sides/tops of the male pins.
Works a lot on games older than this, but you might as well try it since you'll be waiting to fix anyway.

As for replacement, that standard moves you are showing you would need neew ends as stated and a crimper...but how would that be done since the wires run through to another area?

Heck, I would also check the solder job on that opposite site, and the twist in connections. But hey. I like to solder things

Let's hope it sticks!