They are a concern to some teams, sure. I don't know that Boras is lying or swindling the Twins. The teams see his injury history and medicals...past that it is on them to make a judgement on it. Unless he is providing false medical data, it is simply his opinion that it isn't a concern.
It may lower his chance for another long-term deal, but not entirely. Justin Turner got a pretty large contract when he was 34, I believe. If he stays good it may cost him money, but not as much as you would think. You can still get a nice multi year deal in your 30s. Instead of having the next 10+ years of salary based on 2022 salary standards, another contract will get based on salaries over half a decade from now. Compare salaries 6 years ago...pretty wild. It is exactly how the Braun contract extension did end up a big disaster. By the time the 5 year contract ever started salaries boomed. Then by the last few years of that extension $20mil was hardly a large salary and you didn't need to be MVP level to command it.
He may never get a contract to take him to 40 years old....but after this offseason, I don't think any team would have done it anyway. I doubt in 2 years a team would have forgotten all this, see his medicals, and hand him a decade contract. It is fair to wonder if the Mets getting concerned had a domino effect to the other teams. Had he agreed with the Giants/Twins first those teams might not have batted an eye at anything.