If they didn't do an alignment and it's really far off, then it's possible the tires could wear out really fast, like so fast that a Brand new set of tires could be worn to the point of being dangerous in just a few hundred miles. An alignment is a MUST.
Also it's generally a good idea to move that space-saver spare to the rear of the car, The front wheels have the job of accelerating, Doing the most braking, and steering. It's a lot of work they have to do, and that little donut can only handle so much. Especially if you're having alignment issues.
Also on tires they have a wear indicator bar in the tread to give you an idea of when you need new tires...
When that wear bar gets flush with the rest of the tread then you need new tires.
Also tire wear can be read like a book to determine what kind of alignment issues you have. Here's a good graphic for that.
The looseness of the steering could be due to work tie rod ends. If the tire blew and the car spun around, it could have... and probably did wear them out. The mechanic should have told you about everything that needed replacing or repairing.
Getting a second opinion is a great idea and hopefully you can get some more information.
It's too bad it happened but thank god no one else was involved and that you're okay. Good luck!