Mr Hogiewolf, what you describe does sound familiar of a failing thermostat.
Having dealt with this myself only a couple months ago, I share these tests I did.
First verify no obstructions of airflow across the radiators. If in doubt, use hydrogen peroxide, sprayed onto the radiator fins (radiators not hot), let sit keeping wet about ten minutes, then rinse carefully from the rear side, then front side.
Yes, at idle, parked, the temp would increase well above normal. When fans came on, as they should, the airflow when stopped blows forward. With engine idling, temps increasing, place your hand in front of each radiator. Is the air blown on your hand hot, or cool. If the air remains cool, the thermostat stuck, and no heated water is circulating into the radiators.
To validate you do not have an indication issue, disconnect the analog fuel gage and analog temp gage. Doing so will put the indicated display onto the center display. If the center display also indicates high temps, there is no concern of a failed indicator.
Unfortunately, on ours, the temp problem was semi-intermittent. Sometimes fine, other times wanting to overheat.
That said, and not wanting engine damage from overheating, or getting stranded, I purchased a new OEM water pump assembly and installed it.
The price the shop quoted seems accurate. The water pump is a couple bucks under $300 at MSRP. The shop will also charge for new coolant, and several of the OEM type non reusable hose clamps. Shop hours should be between 1 1/2 to 2 hours. This includes draining coolant, obtaining access by removing body panels and the computer. Then swapping the pump and reassembling.
Most folks here that attempted replacing a water pump indicated they would have preferred letting the dealer do the work. Yes, I did the work myself without issue or concern, but then again, my dumb brain finds Spyders to be pretty basic machines.
Entirely up to you how to proceed, but unlike a cut on your finger, this will probably not heal itself.
