Lu, I'm a little rusty on the acceptable failure rates but I do agree with you on the philosophy absolutely. I don't think the FAA and CAA would take any different approach to this. However, I doubt any aircraft tyre has a reliability of 1x10^9, or can given the inevitability of contaminated runways.
Which comes back to RJ's point. No a redesigned tyre almost certainly isn't the sole solution because having established that a tyre failure caused the crash it presumably can again; even if a redesigned tyre reduces the risk, the risk is still there.
In my opinion that's why the tank needs protection too - so that it would take a double failure - tyre+tank protection to cause an accident.
I should qualify this by saying that all of my evidence is from press reports, I have never (sadly) worked on Concorde.
Kegworth was a (737?) inbound to East Midlands. An engine caught fire and the captain chose to shut it down. Unfortunately a very poorly designed engine display, no visibility of the engines + presumably a fair degree of stress caused him to identify the wrong engine, and he shut-down the good engine. Result was a crash just next to the M1, until about 5 years ago you could still see the marks on the bank, left side of the northbound carriageway just before the bridge at Junction (25?).
G