Bimetallic strips for use in temperature protection and circuit breakers are usually sort of cup shaped (like a pop-it) which gives them some snap action when they open and close. That's good for electrical loads because it quickly connects and disconnects, avoiding any arcing. But this seems to not be the case for fluorescent glow starters, the bimetallic strip bends until it just barely makes contact and then immediately starts to bend away. This means that they only make contact for a split second, which might not preheat the tube enough and so they might have to cycle again and again.

Why don't they use cup-shaped bimetallic strips for fluorescent starters? Then they would glow, snap shut for a good long preheat, and then snap open for a nice sharp flyback pulse.

There has to be a good reason why they don't do this, what is it?

Some well-made glow starters actually snap because of momentarily sticking the electrodes, giving a well defined preheat time on a first try.
 

Snappy strip itself has higher stresses, so higher risk of fatigue cracking...

The "ping" sound starters make when starting lamps, does show that the starters do in fact snap action

For how long it stays shorted depends on the shape, thickness, etc of the bimetal strip. In particular, how far it can tension back when cooling down, before the force tears open the micro weld that happened between the contacts due to arcing when it was closing

Some starters have the contact spot of the bimetal strip bent in a specific shape in order to control the surface area of the contact spot, and therefore strength of the weld