Hard wiring is not the way to go - those are Bally style plugs. They are repairable, but it's a lot of trouble - drill out the rivets holding the two pieces of fiberboard to the center metal, then drill a hole through a piece of metal that has been bent to the appropriate shape, then re-rivet.
To replace, you can buy a connector from any parts person - number of female holes doesn't matter as long as it is more than the number of male pins. On the female side of the plug, you desolder one wire at a time and wire it to the new plug.
If you desolder the whole thing at once, you will need to take good photos and label with masking tape or similar.
It's not a huge job once you get into it.
You can replace with Gottlieb if you need to - whatever's handy - but I would replace with Bally. Pin spacing should be the same regardless. I would do that to preserve the look of the interior, but that's me.
As to why otaku thinks Gottlieb/Williams are better - it has to do with how they are made - Gottlieb and Williams had a more solid female connector that was not as prone to breakage. There are no clips to catch with a stray piece of fabric, and they are easier to clean.
You can pick up female jones plugs from this thread or certain sellers on ebay or a bingo guy named Joe. PM me if you want his info. Depending on the size needed, they are not necessarily cheap (they're no longer made). They are sold based on the total number of connections - so when you say four, do you mean two wires on each side or do you mean four on one side? The selector plugs could be deceptive if you're just getting into it, since you only use 1/2 at a time.
Most often the plugs are sold in pairs.
You can also 'just play' with the alligator clip. But I would replace. Prevents a clip from falling off and making connection somewhere else. Especially on a backboard connector which is also fighting gravity.