However, I'm unsure which type of PhD lab would provide better preparation: either a problem-first lab (e.g., like Yet-Ming Chiang at MIT) or a technology-first lab (e.g., your typical academic lab that focuses on novel science without a predetermined application, and where commercialization happens largely by chance — when a technology happens to have a valuable market application).
I've looked at advice online from successful hard-tech entrepreneurs, but the answers are conflicting. Some say to work backwards from a problem, but others argue that problem-first approaches often don't work since deep tech is inherently different: you often can't force a scientific breakthrough for a predetermined problem. Instead, they argue that most hard-tech companies are only founded because somebody made a scientific breakthrough and realized afterwards that there might be a commercial application. Indeed, the VC firm Pillar VC says that most deep-tech companies they know were "technology-first."
With that in mind, does anybody have any advice on which type of PhD lab to join?
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.