Well something was going very wrong for a very long time.
And what was (really) done about it? Below from Air France Accident Report.
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Research was undertaken to find incidents which had involved tyres or landing gear on the Concorde since its entry into service. The information collected to establish the list of events came from the archives of EADS, Air France, British Airways, BEA, AAIB, DGAC, CAA and Dunlop.
The list in the appendix shows information from events coming from at least two different sources or for which reports or detailed information exist.
In the list, there are fifty-seven cases of tyre bursts/deflations, thirty for the Air France fleet and twenty-seven for British Airways :
Twelve of these events had structural consequences on the wings and/or the tanks, of which six led to penetration of the tanks.
Nineteen of the tyre bursts/deflations were caused by foreign objects.
Twenty-two events occurred during takeoff.
Only one case of tank penetration by a piece of tyre was noted.
None of the events showed any rupture of a tank or a fire, whether leading or not to a significant simultaneous loss of power on two engines.