Well, a five year old stat might be getting sticky....
...or it might not.
But, this is around the time I would be recommending a new one.
item # 5 in the link below
I also recommend taking off the water jacket cover (item # 8) and inspecting/cleaning out the jacket. Especially for this symptom.
That gives you the opportunity change out the sacrificial anode (item # 1) and it's mounting screw (item #12) (don't scrimp on that screw, they corrode and will break off causing you more than $2 worth of misery)
Check/clean the flush valve components (items # 2, # 3 and #4). They can typically be reused but I prefer to change them out...the spring at least.
If you choose to do this chore, buy new seals (items #1 and #9) as they don't always cooperate when being reused.
Doing this bit of maintenance may not actually resolve the issue but, as I said, it is time.
One way or the other it will eliminate the stat and crud in the water jacket as being the cause.
You mentioned sensor before and, yes, occasionally they fail and could cause this problem.
But I don't care for the "test" that has you unplugging it and running the outboard because doing that leaves the engine unprotected in the event that it IS actually overheating and you didn't detect it.
For $25 bucks I would just add one, with seal, to my parts order.
Items # 8 and 15 in the link below.
Good luck and please let us know how you get on.