You wouldn't think that driving on a flat tyre would affect steering, unless it had been potholed or something, and that would in all likelihood damage the rim.
Is the new tyre a significantly different tread pattern, or the same tyre type? I have had some odd effects from mixed tyres on the same axle, especially directional patterns with symmetric or asymmetric.
Check that it is actually pulling to the left, and not just the camber of the road, by driving it down the crown of the road. Occasionally, some people only notice something odd when something else goes wrong, and connect the two.
If it is still pulling to the left, there are not many other options left, apart from trying someone else to check the tracking - it's amazing how many garages are poor at steering alignment.