stats on these tend to fail open, so you would not get the symptoms you describe..
I think dropping the coolant and replacing is a good starting point to diagnose your issue, but there are basically two things that could be happening..
1) the system has a leak somewhere that is resulting in the system not being pressurised, this would cause the coolant to kettle at lower temps, as most cars coolant systems run at around 25 psi, each PSI above atmospheric increases the boiling point of the coolant by roughly 1 degree. so when you are at 90 on the temp dial you should be actually around 35 degree lower than boiling point...if its boiling earlier then its not pressurised or being over pressurised by exhaust gases.
you can pressure test the system with a pump and schrader valve/modified rad cap with a pressure gauge to cheaply make something like this...
with the egine not running its easier to see/hear any leaks...
the other obvious is that you have a headgasket breach which is pressurising the coolant even before you get to temperature..I would run the car up to temp with the cap off and look for bubbles in the expansion tank, its not definitive but if the system is bled properly there should be no gas of any kind in there at all, and bubbles must be coming from somewhere...
as a starting point do what you are planning but also change the rad cap, and visually inspect all the pipes and hoses, it's amazing how often its a banjaxed o ring in the rad cap causes the system not to pressurize.
I would also put some cardboard under the car to help you pinpoint exactly where any leak is coming from, but TBH it sounds like if you have one its small and venting to air as vapour.
hope that helps,