If it's overheating that fast after the idle comes down, something is not right. The three things that come to mind are the thermostat, air pocket in the cooling system and the sensor as being the least likely as I have not heard of any failures of these although anything is possible. If it's not too expensive, could be a quick way to eliminate that.
When you fill the system, do you remove the bleed plug in the top pipe? It's a small rubber plug with a spring clip holding it.
I've had mine idle for 10-15 minutes to when the fan kicks in and off in a 20C garage. The only time I ever overheated was when I was running in very low snow and the radiator couldn't keep up. Let it idle for a bit, came back down, no problem.
No diss taken. We realize you inherited a basket case so there are a lot of unknowns here.