As with a lot of things, the Challenger Accident was far more complex than just that.
Yes - the SRB O-Ring design was not made to withstand the sustained low temperatures and then launch/hold the joint seal. HOWEVER - that was outside the design specifications, so blaming Morton-Thiokol *solely* for this was not the way the Rogers Commission report went down. Sure, the joint failure was the root cause of the *physical* incident, but it was the personnel issues - especially cultural and managerial - that were every bit to blame up and down the NASA chain.
Thiokol was a convenient public scapegoat for the media, and when Richard Feynman dipped an o-ring segment into a glass of ice water - and CNN covered it live (https://www.youtube.com/watch?v=6TInWPDJhjU) - that's practically all the subsequent coverage centered on. Internal to NASA, though - we knew who the decision-makers were, and they weren't in those roles for much longer after that (voluntarily or not).
We went through a lot of soul-searching, including lots of training and process improvements - did some good "cockpit resource management" and "breaking the error chain" training during that down time. Sadly ... some of those errors crept back into the NASA culture for poor decisions made that led to the loss of Columbia less than 20 years later.
If you want to really dive into it, the report is online here (PDF). I still have my official bound copy that pretty much all of us had as a reminder of the failures of *people* as well as *technology* that led to that horrible day.