When an engine is subject to a precautionary shutdown should the continued progress of the flight be on the basis that this is a failed engine
In a twin I'd say yes and land at the nearest suitable as, I'm sure would everyone.
As you say, in dire straits, with second donk acting up you may HAVE to try a restart. Command decision depending upon reason for first shutdown.
Flying one of them there single turbo-props strikes me as a situation where the captain is going to have to nurse the engine if there's a problem. I'd a JP donk go nuts due severe icing but just closed the throttle and kept it in reserve for the subsequent deadstick. It worked for the taxi in but was knackered.