I have a PhD in mechanical engineering.

I would back up and instead ask whether getting a PhD is a good idea. Unfortunately, academia is a minefield. PhD students are largely cheap labor. Getting a PhD can be a valuable apprenticeship, but often it's abusive and poorly paid. You might nominally get some freedom, but the grant funding wants you to do something you might not care for. The opportunity cost of a PhD is huge. You should consider as an alternative taking a more conventional job, maybe a part time one, and doing science on the side to figure out how to start your business. A part-time engineer making $50K/year is getting a much better deal than the vast majority of PhD students.

The problem-first vs. science-first framing sounds good on paper, but it would be difficult to accurately determine whether a particular lab has either focus ahead of time. What you see from the outside is mostly marketing. Most labs are really neither and instead are publication-first. Publications are the currency valued in academia.