1) Not likely. Transformers are solid state. They have no moving parts as they are just loops of wire and iron. They can go bad when they get too hot by melting internal insulation but I've never seen one magically repair itself.
2) I'm not sure if the transformer is a replaceable component.
1. Check transformer output with a meter. Its probably 120 in and 24 out.
2. Monitor boiler when it calls for heat and see where the train of events stops.
3. If the aquastat is in fact the problem, check the contacts on the relay and make sure they are clean. Low voltage, open air relays in dusty environments are a recipe for trouble.
4. Double check all relevent interlocks like water level, high limit, flue damper, flame proving, etc.
When the boiler calls for heat and fails to start, does the little switching relay activate (pull itself in)??? Or does it fail to even move on its own unless you do it manually?
If it doesn't even move then you can ignore cleaning the contacts on that relay. If it does click and try to activate, it might be dirty contacts.
The problem with calling a plumber is that they are not electricians and you have an electrical problem, not a leaky pipe.