When I first started reading this, I thought I had a solution. Actually, I do, but it seems to be more than you are looking for.
Most automotive temp switches are going to be within a fairly broad range. Most of the time its not cost effective for them to make them "tighter", and the system they need to control usually has a fairly broad "good" range.
If you want more accuracy, you need to move out of automotive and into general electronic controls. PTC or thermocouple temp sensors and controls are very accurate, repeatable, and can be set to whatever temp and dead band you want. Plus, would give you a readout of the actual temp you are measuring.
I recently bought a four channel K type thermocouple system for about $35. I'm going to install it on my tractor to monitor some items.
It doesn't have control capability, but I don't need that. I just want to be able to monitor four temps in real time.
Right now there are lots of cheap units out there. Some of them might be too cheap to trust your cooling system to. But since you have manual on available, you could add a single overtemp switch that would warn of a control failure.
Just a few seconds of checking found these. Degree C only, but it would be hard to beat the cost....
Or this one. For some reason, even though they make really cheap units, I have had great luck with Drok.
The sensor leads seem too short, but since they are NTC, I suppose you could lengthen them. Might cause a slight offset in the reading, but you could easily compensate for it.
If those won't work, there are lots of others out there. Thermocouples would give you a very wide range and can be extended with the correct wire.