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.

In my totally uneducated opinion as a founder working on winding down my software company that can't differentiate anymore and considering finding someone in hard tech to partner with, I think problem first is a mistake. Problem first is appropriate for a business, but for a lab, I think it should be organized around the art of possible, deep specialization and capabilities first. You never know what valuable things will come out of that process. Once you have something promising, you can match it to problems and make a business case working backwards from you. Generally startups are the opposite, you want to organize them around a customer and a problem, but fundamental R&D != startups.

First, not clear PhD is really needed, speaking as a PhD. Second, if you do a PhD, make sure your professor is ok with summer internship type things.

I think on net it's valuable to become a good problem solver and thorough thinker. Domain expertise is also good, especially if broadly applicable. The most important things for success in PhD regardless of desired path is 1) good advisor, background check them, ask current and former students 2) loving the problem and the problem having some reasonable chance of progressing 3) funding.

Now specifically to your question about the flavor of the lab, my gut says favor the one that's more about hard problems and good methodology and innovative "tool" use(especially ai) regardless of domain. I also would lean against something trying to prove a particular technology with the pi maybe thinking they can spin off a company. That's not the best use of PhD time unless the PI has a spectacular track record of startups and connections. Almost always PIs and labs don't exactly mimic the start up scene. Basically the idea that just one good invention that's technically a better solution than everything out there is almost never enough.

I guess my rambling point is that if you're going to do a PhD, do it for the skill set and differentiation. If you treat it as prep for a specific start up like a tech focused lab, you would be better off joining a startup in that sector.

I don't know what the VC is saying, I don't why a deep tech company wouldn't be technology first?

Re-reading again, sorry for the rambling: don't count on one particular approach to one technology yielding a good startup. There are exceptions, of course, especially if you're aiming directly for acquisition. But generally, a good tech startup isn't about a brand new technology, it's about integrating technologies and adapting academic knowledge practically to industry problems. Especially since you seem interested in heavy industry (where I went), look for places where no one knows anything about the tech and knowledge you bring, talk to the them, especially the operators, figure out how they really actually do shit now. Then reimagine it using your approach, always talking with actual operators.

  Thinking out loud... I would say probably a good filter right now when AI in the near future makes the dirty translation/onboarding layers easier, look for places where startups have recently failed to penetrate an industry and take the second mover advantage.

Note I'm thinking mostly of heavy and dirty industries, primed by your message.

Edit, don't know why that thinking out loud section is formatted like that, not intentional.

Some good advice I once heard can't remember from who: if you're really smart, don't go where everyone else is really smart. But the caveat there is your personality must match, existing heavy industries have seen their share of arrogant geniuses who spend no time trying to understand how things actually work. Approach people openly and with curiosity, like they have something to teach you, because they do.

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