+1 to everything T3sl4co1l posted.
It's ok to change just the FET and try again, though. If the thing doesn't work anymore, then work your way back and hope you can find the driver and other parts.
I've done a little reading online to get a basic understanding of what a MOSFET is, I don't quite understand what it does,
There are a lot of specs on a FET, and most of them don't matter for this use. In this case, it's just a switch to turn the saw on/off. The main reason a physical thumb switch doesn't just turn the saw on/off, directly, is the circuit board has to also be able to cut power when the battery runs low to prevent the battery from being a single use item. And the circuit probably also switches off the power in case the motor draws too much current (like what would happen if you were to stall the blade in something), which could irreversibly damage the battery and/or the motor. For the circuit to do that, it uses the FET as a switch, since it can't flip a thumb switch like you can.
If you get the idea that you could just trash the board and replace it with a physical thumb on/off thumbswitch and some common sense, I didn't say it. It would be real easy to overdraw the battery and damage it. It would be somewhat feasible to make your own protection circuit with low voltage cutout and reset as a learning experience, but it's highly unlikely the saw itself is worth that much time in the end.
The two parts I have listed are made by Infineon, are there other brands I should be looking at? I didn't know how to filter down my options, because there are a lot of numbers and information on these MOSFETs and I really don't know what most of it means.
Brand names don't really matter, so long as you buy it from a regular component retailer and not eBay or Alibaba. Most of the specs don't matter to you other than
1. Correct package. Same package as the one you removed from the board, whether that's a TO220 or Dpak or whatever it is.
2. Rds on, resistance between drain and source.
3. Max voltage of 40V. Don't go too much lower. I bet 30V would work. You could go higher, but it would probably results in worse specs for no reason. 30-40V is fine.
4. Peak current draw. The peak is going to be pretty high in any inductive load, like this. Stick with 750A peak minimum.
The constant/static current draw is also important, but take that with a grain of salt. What you really care about is the package is the same one that came off the PCB and that the Rds is about the same as the original. If the constant/static current draw is rated higher than the original part, but the Rds is also higher, then it is not going to be "better" and it may die, prematurely.
You could go a bit smaller/lower on the Rds than the original without too much issue, but that might affect the cutout voltages of the protection circuitry.
Most of the other specs shouldn't really matter for this purpose. I'd ignore every other spec and just sort by price.
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