The phrase “work well under pressure” is something that I consider to be a red flag in a professional environment. My company builds a database that is used as the backend of business critical systems. If something breaks, there is a need to fix it. It costs money (sometimes a lot of money) for every minute of downtime.
Under such a scenario, I absolutely want the people handling the issue to remain calm, collected and analytical. In such a case, being able to work well under pressure is a huge benefit.
That is not how this term is typically used, however. The typical manner you’ll hear this phrase is to refer to the usual working environment. For example, working under time pressure to deliver certain functionality. That sort of pressure is toxic over time.
Excess stress is a well known contributor to health issues (mental and physical ones), it will cause you to make mistakes and it adds frictions all around.
From my perspective, the ability to work well under pressure is an absolutely important quality, which should be hoarded. You may need to utilize this ability in order to deal with a blocking customer issue, but should be careful not to spend that on non-critical stuff.
And by definition, most things are not critical. If everything is critical, you have a different problem.
That means that part of the task of the manager is to identify the places where pressure is applied and remove that. In the context of software, that may be delaying a release date or removing features to reduce the amount of work.
When working with technology, the most valuable asset you have is the people and the knowledge they have. And one of the easiest ways to lose that is to burn the candle at both ends. You get more light, sure, but you also get no candle.