This is my take on your situation.
1. the motor is not over heating, thats good.
lets look at how the water flow works,
impeller in outdrive supplies water to engine,
engine water pump circulates the water in the engine
thermostat controlls the temp to set point determined by the thermostat.
WHat influences the water flow,
At idle the raw water is pumped up to the thermo housing, the housing has two bypass ports that allow raw water to flow to the elbows (typically) to cool the exhaust exiting the manifolds.
The thermostat stays closed until sufficient temp is reached then it cycles to maintain that temp.
A normal system will stay at 150-160 all the time and when running at higher rpms it will drop back 10-20 degrees due to cold water rushing in then it will catch up and stabilize at typically a higher temp, 160-180 depending on thermostat rating. When you slow down and come back to idle it may go higher for a short time but will come down to normal 150-160,
So in my opinion there are several factors that can change the normal operation.
lack of water in, due to bad impeller or torn gaskets at impeller housing ect etc.
restriction going out such as rust scale etc etc.
both of the above would cause a over temp situation.
You are "seeing" a under temp situation.
That conditon shows what happens when there is NO thermostat in the housing. The water flow is less at idle so the temp stabilizes. as rpms increase the water flow/volume does also and therefore cools the engine down as there is no thermostat to controll/maintain the temp.
So assuming your water in is good and your water out is good, no restrictions or issues at impeller then the issue has to be in the thermostat housing.
Either the thermostat is installed wrong, or the housing is bad, ie rusted and the bypass ports are bigger than normal or other areas inside may be open/larger than normal due to corrosion. Or maybe just a bad thermostat. uncommon but can happen.
the last thing that could be wrong would be the sending unit itself.
The sending unit is no more than a variable resistor and the change in resistance should be linear. What could happen is the sender get warm at idle and displays the temp you see but when colder water hits it as rpms increase and the temp gets cooler the sender drops but does not recover untill the engine stabilizes at idle ( long shot and I may be a bit over caffiened this morning)
Other and last possibility is bad ground at sending unit to manifold. the threads of the sender must be clean and no sealent of any kind used as this is the grouind connection for the sender.
Best I can come up with for now.........hope it helps......
I myself, my temp guage rarley goes over 120 but I am too lazy to figure out why............lol
Most likely in my case it is the sender, its 23 years old.........