Choosing an Engineering PhD Lab: should students start with a “big question” in mind?

I’m curious how mechanical + electrical engineering PhD students should think about choosing a lab. Specifically, my goal is to maximize long-term scientific impact, but I’m not committed to a single problem. That said, there are several big problems I care about (e.g., energy storage, desalination, etc.) and would like to tackle throughout my career.

For this reason, I’m unsure whether it is better to join:

1. A problem-focused lab whose primary focus is to work directly on one of those problems (e.g., an electrochemical battery research or desalination lab).

2. A technology-focused lab (e.g., meta materials, novel membranes) whose technology possibly could be impactful for the problems I am interested in (but chance is low). However, in general, the technology is versatile and could be applied to many problems in the future.

3. A technology-focused lab also working on a versatile technology, but not obviously relevant to the problems I care most about.

**When choosing a PhD advisor, should students already have a few “big questions” in mind and let those largely drive their decision (i.e., favor a problem-focused lab)? Or are there good reasons to choose labs 2) or 3) as well (and if so, why)?**

Also, for comparison, how does the answer differ in industry? Is the usual advice for junior engineers to expand their skillset as much as possible early on, or to specialize around a particular problem?

Would greatly appreciate any insights, especially from professors or researchers who have advised PhD students.