Strong work...yes your thermostat is 100% toast.
After my replacement of the thermostat it runs consistently 89-91 C

I paid an Indy to change my Thermostat with a bmw one from ecs tuning for around $450. He only charged me 0.25/hr to change the mechanical water pump while he was in there as well. He recommended this and I’m glad I got it done. 161,500 miles and counting.

Here is my notes from my repair.

BMW repairs

$56 thermostat with o-rings

$137 water pump plus gasket
Coolant

Labor
4.5 hours
2 gallons of coolant