Why did the space shuttle use bias-ply tyres instead of radials?
Simply put, for the weight bearing factor bias-ply is a lighter tire, and when building a spaceship weight is the top concern. 

The reason they were thin and single use also had to do with weight, according to NASA:


  Weight: Since weight is of extreme importance, the tires are made with
  a minimum amount of tread to conserve weight, allowing for larger
  payloads. A few pounds may not seem to make much difference, but when
  you add up all of the ways to decrease weight throughout the shuttle
  it can have a significant impact.


When it comes to weight bearing capacity: 


  Load-Carrying Capacity
  
  Advantage: Bias Ply
  
  Due to their multiple layers, bias ply tires generally have the
  advantage when it comes to load-carrying capacity. Because there are a
  uniform number of plies to support the weight of a vehicle in both the
  tread face and the sidewall, bias ply tires are generally better when
  hauling heavy loads. This is why trailer tires, tractor tires, and
  heavy equipment tires are often bias ply; they can better support
  heavy loads. Radials are available with high load ratings, but the
  weight-carrying capacity requires adding many more plies to a radial
  tire, making them very stiff.


The important line here being "adding many more plies" which will also make the tire heavier (reference above note on weight). It costs in the neighborhood of $10,000 to put 1 lb into space, so if you can save a few pounds on each tire, it does not matter that they cost \$5,660 each; you save by replacing them every time if they are indeed lighter. 


  even a single MLG tyre failure would significantly impair the vehicle’s braking capability, which is most emphatically not a good
  thing for something going as fast as a landing shuttle orbiter.


The shuttle has no issue coming to a full stop with a blowout as was the case with the shuttle Discovery in 1985 during the landing of STS-51-D.