I'd advise giving your current lead every benefit of the doubt to figure out if the problem is truly a mismatch of their skills to leadership or if there's something happening beyond your team. In the latter case you may not want the job...

Your lead was around when hiring was tough. Moving from IC to eng lead is the clearest growth path for an IC that I can think of, so I'm not sure that aligns with their rationale of "never be[ing] able to grow within the company." Knowing nothing about this person, I think there's a chance there's information you don't have here (not your fault, you weren't around to get it!). "I'm a better IC than lead" is a solid political answer--and it's true for many/most? of us. But it's also a _safe_ answer.

Why were jobs unattractive to external candidates during the time he was hesitant to take on engineering leadership? If the answer is "no one wants to relocate to our un-hip city" that's understandable. Most other reasons? Dig. Keep in mind this person, regardless of skills fit or raw coding prowess, has been exposed to different information and experiences at the company than you have. They have insight into culture and middle management that you might not.

Part 2, to actually answer "What do I do?" - OK, so you gathered data and the lead is indeed simply not a good fit in their role. Every other team around you is firing on all cylinders - in fact they whisper about your team's localized problems. Everyone on your team would be supportive of a change. What then?

If you're going to go to your direct manager or skip-level and ask for a bigger role, make sure you have a good argument from their perspective of why. I'm sure this is in the 80,000 negotiating books that are out there that I haven't read a single one of. Doesn't matter how you see the world, it matters how they do.

Knowing nothing about how awesome you're doing in your current role: what have you done for the company recently? Do skip-level managers know you? Do you have a reputation as a problem-finder and fixer who can work across teams and just gets things done? Have you been owning projects mid-level managers have heard of, even if you're not team lead? Are you "the hero of GDPR compliance" or even "person who wrote the extension that fixes my email nit?" If so, awesome, say that. "I think I'm a better candidate than person X" and "I had a similar role at my last company" are less likely to meet the bar.

If your team's problems are affecting your personal output, then unfortunately that's going to make your case a little more difficult :(. If that's true, try to find a role that lets you show IC skills at the "very good" level (need not be "incredible") while also adding an informal leadership merit badges here or there and getting at least some reputation beyond your team. That's what I've seen in pretty much every case of IC getting leadership responsibilities (which isn't a formal promotion at our company).

Just get enough info in advance to be 100% sure you want the job :).

Semi-related concern if you did take on this team's leadership: are there coworkers that think they could do better than you if you did get promoted? That can cause some people to find other opportunities. Make sure you'd be ok with that.