I don't understand how changing the thermostat in the airbox helps a cold start problem.

That thermostat selects preheated air, but that air won't be preheated until the engine has run for a minute, or so -- until the exhaust manifold warms up. Until then, that intake source is just as cold as the direct source.

You must start the engine (cold) and run it before that air starts to warm. So, even with a new t-stat, that air is at first very cold.
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Don Foster (near Cape Cod, MA)