No doubt the Concorde crew had a few moments to realize that the blown tire might be more serious than just a blown tire.
The question of the thread didn't ask about a crew that took off and had no idea a tire failed. They knew. They were aware of the problem on arrival and briefed the passengers. The question isn't about a problem that's unknown. It's about a known problem and the decision tree that follows.
The MD-11 has a kevlar plate mounted on the gear, above the center wheels, which sits aft of them when the gear is retracted. The purpose of that plate is to protect hydraulic lines and the reversible motor pumps from damage or destruction in the event of a wheel failure. Without them, a tire failure on the ground or in the well could very easily result in another sioux city event. Loss of all three systems.
A tire may have punctured the underside of the aircraft, penetrated a fuel cell, damaged hydraulics. To know of a problem, have a runway nearby and the chance to return before fuel becomes fire, before a hydraulic loss occurs, or any number of other problems develop, and not take advantage of that runway may be to throw away the only shot.
The UPS 6 crew that departed Dubai and got an inflight fire, was abeam Doha when the airplane began to burn. They could have put in at Doha and survived; the timeline would have allowed it. Instead, they tried to turn back to Dubai and didn't make it. The notion of continuing along one's way after that tire failure (it's just a tire...what could go wrong, right?) sounds innocuous enough. What's that folks say about assumptions?