They surely made a mistake, but I don't believe it is what you are accusing them of here.
They had a long history of debris strikes in previous launches. They did not know that there was catastrophic damage to the wing's heat shield. They noticed the strike after the shuttle was in orbit. They elected not to disrupt the schedule of the crew or request additional observation resources (i.e. ground telescopes) to inspect it, and instead, they did some mathematics modeling of the strike which ended up being inconclusive, so they chose not to spend more time and money investigate further.
That is not the same as thinking they knew there was a problem and the astronauts were doomed but they decided not to even bother trying to rescue them.
A lot of the "didn't even try" is buried in your "elected not to" phrasing. Yes, they didn't check out the damage visually, which is another way of saying they didn't even try to assess the damage. At least part of that decision seems to have been driven by the fatalistic idea that a rescue was impossible and so it didn't really matter.
Also, I reach the opposite conclusion you do from "a long history of debris strikes". To me, this says that they absolutely knew there was a problem, and had known it for a very long time, but merely got used to it and dismissed it due to the ridiculous idea that it's OK to ignore if it hasn't killed anybody in the past. That history makes them look worse, not better. Same thing happened with Challenger and the O-ring leaks. The leaks were a serious problem and smart people said as much, but the concerns were dismissed on the basis that previous flights hadn't exploded so the leaks must not be serious.
Agreed. From wikipedia - "Engineers made three separate requests for Department of Defense (DOD) imaging of the shuttle in orbit to more precisely determine damage. While the images were not guaranteed to show the damage, the capability existed for imaging of sufficient resolution to provide meaningful examination. NASA management did not honor the requests and in some cases intervened to stop the DOD from assisting."
Judging from the article, a rescue mission of the kind described here would have effectively been impossible. Even getting to the point where it seemed like a good idea to try would have, in the overwhelming probability, still ended with the loss of at least one shuttle and its crew.
They had a long history of debris strikes in previous launches. They did not know that there was catastrophic damage to the wing's heat shield. They noticed the strike after the shuttle was in orbit. They elected not to disrupt the schedule of the crew or request additional observation resources (i.e. ground telescopes) to inspect it, and instead, they did some mathematics modeling of the strike which ended up being inconclusive, so they chose not to spend more time and money investigate further.
That is not the same as thinking they knew there was a problem and the astronauts were doomed but they decided not to even bother trying to rescue them.