You are correct that the devil is in the details, as they say.
One first needs to know what PSI the oil is generating.
Then one must find a tire that can withstand that pressure. Maybe fighter aircraft tires.
Then one must find a tire with those characteristics that is the correct diameter.
Then, assuming that one tire will not do it, but that the tires must be stacked on the core pipe, how a stack of tires maybe 50 or 100 ft high will all get inflated at one time.
Oh, yes. One has to have an air compressor that will do the job in such a manner that all will inflate simultaneously or the inflated tire will be crushed by the oil pressure before the next one can inflate to back it up.
And then there is the small matter of how you get a bunch of limp uninflated tires to withstand the huge pressures and rate of oil flow long enough to get them down in the pipe.
The idea is probably a good one, but not using tires.
In the length of time spent and the predicted time frame, I would think that someone could manufacture an inflatable device that would start out narrower than the pipe, designed to withstand the pressures and inflatable from one end.
Sort of like a huge tough hot dog with internal cells around a pipe to give it a spine for insertion.
But then the Russians have used an atomic explosive device six times and it worked, so if stopping the leak was really the number one concern, that would have been done already.
Better to have the oil spill all over the TV than the Blagojevich trial fall out
Something like the airbags that are used on the suspension on 18 wheelers? Like you say, though ,getting them in the well, and then inflating them would be a problem.