I have an older Craftsman Riding mower that I have to wiggle the wire connector to get it to start. The ignition switch is working properly (I checked it with a multimeter) so I think the issue has to be in the wiring connector. Is there any reason why I can't remove the wiring connector and replace the connections using female disconnects? A new wiring harness is $70 and I am looking for an alternative.
Yes, they went out of their way to use a standard type part so you could do that.
Those connectors do have 3 different widths, so watch out what you are buying.
When I am faced with this kind of situation I evaluate time, money and the likelihood I will be doing the job again and also the cost of a new system.
That said $70 is your basis + your installation labor + you might need to replace the female connection and wiring.
So that said one option is to replace the connectors Male / Female with a better quality and more standard available connector such as the below link.
These connectors are rugged and the MS 5015 link shows units acceptable for Military Ground Equipment , there are other higher quality units used in Military Aircraft as well - but you don't need that.