It's difficult to be certain from the images you supplied, however the assembly with the spring inside the end cover that has been removed looks like it could be the centrifugal (stationary) start switch. You can probably confirm that it is a spring-loaded switch by applying pressure on it at the point where the shaft passes through it. If it responds with a "click" action then it is definitely a switch, and/or you could confirm by checking for continuity across those wires.
The assembly protruding from the shaft with the spring looks like it is the centrifugal weight that activates the switch, although again it's difficult to see how it operates. If you "lift" that disc away from the shaft (i.e. to compress the spring) and that causes the disc sitting on the shaft to move towards the end of the shaft (i.e. towards the switch in the end cover) then that would confirm that it is the centrifugal switch start mechanism.
If that's not the case then another possibility is if the shaft moves axially when the motor is assembled (i.e. if you can push the shaft in) to activate the switch then it would appear that the motor is designed to run only when pressure is applied to the end of the shaft. In that case I would assume that the machine has a mechanism to press the shaft in when the operator applies pressure to some part of the machine, either as a convenience to the operator or as a safety feature (and perhaps both).
Checking the continuity between those wires should also give confirmation of what it is. If there is continuity which is broken when pressure is applied then it is a "start" (centrifugal/stationary) switch. If it is open but gets continuity when pressure is applied then it is an operation/safety switch.
To answer your question:- no it's not likely that you would be able to run the motor without re-connecting those broken wires. If the switch is part of the starting mechanism, then best case is that you may be able to manually spin the shaft to get it moving before applying power (I wouldn't risk doing it after power is applied), assuming that the shaft spins freely enough to keep rotating long enough for you to then get power to it. That would only serve to see if the motor otherwise still operates, of course, since it still needs to be repaired before you can fit it back into the machine. If it turns out to be an operation/safety switch then of course it will need to be reconnected and also pressure applied to the shaft (I wouldn't risk trying to do that with the motor connected to power).
If you do decide to attempt a repair, pay careful attention to where those wires end up as you reassemble the motor as they will have to stay clear of that assembly on the shaft which will of course be rotating when the motor operates.