Depends what XBOX-1 you have. If you have a Rev1.1 then they can overheat the GPU (they have a smaller heatsink with a fan). If you have a 1.2+ then the only reason you can get overheating shut down is if (A)The fan is clogged / disabled / spun down or (B)The hard drive you've added puts out a LOT of heat.
It is easy to replace the case fan in the XBOX1 with a silent one which also has a much higher CF/M rating.
IMO this is the best fan for an XBOX-1
Add this gasket and even on full (7v) fan speed you'll never hear it.
It fits without cutting the case and uses the 7v fan connector which means you still have fan speed control. It does not fit the fan connector slots so you need to hold it in place in the XBOX using cable ties.
This is not a fiddly repair and you cannot break anything by performing it so I'd recommend you did this yourself.