Why aren't flights continued after losing a tire on rotation?
Blown tire - can't raise gear, shredded tire will not fit in the bay.    -> 

Gear down - much lower maximum speed and lots of additional drag -> 

Both of those mean very poor fuel economy -> 

Very poor fuel economy means not enough fuel to reach destination

So that's off the table.  Given that, there's no question that it's better to follow the safest course... And you had an extremely high pressure tire just explode, a couple feet away from the airplane's skin.   

Especially when that brings you back to a field that is in fact an operating base for that airline, and is likely to have inspection and repair services, and may have a spare airplane.  I could see a Southwest flight out of SJC diverting to nearby OAK, since OAK is a main operating base for Southwest and will have an easier time fixing or replacing the aircraft.