Since he manually pushed in the contactor, aka starter, and it didn't run, that would mean the contacts in the starter didn't make. By jumping around the contacts, and the motor ran, it just fortifies that train of thought.
The float switch would only supply power to activate, aka pull in, the contactor. In other words, the float switch is control power to the starter.
Whether the float switch is good or bad, by manually pushing in the starter or jumping around the starter, you have by-passed the float.
If its running ok now, I suspect the starter is going bad or had trash on the contact points. We have an ant that loves to get into electrical stuff and that is a common problem with starters, pressure switches, etc. in this part of the country.
