It may also be worthwhile having your radiator pressure tested.
Two weeks ago I had a similar problem with our other car. It was dripping fluid reasonably slowly but left a pool of coolant when I stopped (after first noticing).
I pulled the radiator out - the hoses were in good shape so I had the radiator tested. One of the end tanks had a hairline crack and coolant was leaking out. Bizarrely, it wasn't doing it all the time either. Might be worth a look.
Have you checked your coolant level in the radiator (not the overflow reservoir) as the rapid temp rise may not be a faulty thermostat. If your coolant level is low, once the thermostat begins to open, there may such a low volume of coolant in the block/radiator that it heats very quickly, yet there is enough to maintain safe operating temp.
Think about how quickly the kettle boils when there is only a little water inside compared to when it is full. Assume the engine is the kettle element - it is a constant source of heat and the smaller volume means it will heat quicker.
I'm not saying it is one way or the other, but thought it was something to consider.