Just a personal opinion
Some years ago a captain performed a cautionary in flight shut down (IFSD) of an engine on a 320, in result of an ENG 1+2 FADEC FAULT spurious message.
As a fleet manager, I asked him why did he do that, instead of operating normally or at least kept the engine running at idle, as apparently, full engine control was available.
The reply was: “could not be sure of engine condition”.
To me, shutting down an engine when not called by ECAM or QRH procedure, must be based on very solid grounds, and substantiated on erratic or loss of governing capability.
I’ve always interpreted the “lack” of precise procedural action on this case, related to the multitude of possibilities left over for the extend possibilities of the failure itself.
So, to resume my point: if the engine speed can be commanded and no erratic behaviour is present, continue normal operation. If not, evaluate the possibility to operate at idle, if any abnormal is noted, then perform the precautionary IFST,in accordance with the manufactor procedure.
Cheers, C.