Mostly.
What you have described is exactly the behavior of the cooling system. However, it's also one closing in on the heat exchange capability of the radiator. The thermostat will stay closed until 180, begin to (slowly) open and ideally reach an equilibrium point and stabilize.
However, if the radiator coolant begins to heat up the engine will slowly gain heat. in response the thermostat opens further. If the cycle keeps happening (it's a dynamic constant adjustment) at some point it's clear the radiator temperature isn't cool enough to reduce the engine heat (which is why the engine temp keeps rising despite the thermostat continuing to open more and more).
What happens then is the thermostat is fully open at roughly 203f. I don't know your machine, but most of them have a "fan on" command that's about 205f, wiht a "fan off" command about 195f. The fan forces more air through the radiator, the coolant temp drops, and the engine begins to cool down.
Wash rinse, and repeat. Exactly what you're seeing.
And there isn't anything wrong with the fan being needed. You were after all going slowly, so it had no natural air flow. That's exactly the fan's job: replace air flow at slow speed.
However, you needed the fan doing gentle riding on an 80 degree day? Personally that would drive me nuts and I'd be increasing my native cooling capacity. Mind you there isn't anything "wrong" with what's it's doing. But personally I'd be:
Washing my radiator (from the back)
Replacing my coolant with G-05 mixed to 40% anti-freeze and 60% RO water.
Adding in Rislone's "Hyper-Cool" while i was refilling it.
Following the directions for your machine re: air bubble burp.
Why? Because what the hell is it gonna do when you're towing a buddy home on a 95f day? Melt? Not on my ass anyway.
Let me know if you need to know more about why the coolant combination I've explained.