How to grow as an engineering leader with mid-level hands-on experience?

I'm largely in the same boat. I was part of the founding team, architected our systems, wrote a good part of it, and eventually look over leadership of the engineering and product teams as the CEO had to focus on more external matters. I played the SE role on any meaningful call, worked closely with marketing, and even did sales directly a couple of times landing us an LOI at a critical time.

We had an "ok" exit, not enough to never work again by any means but it put some decent amount of money in the bank.

After riding out the acquisition for a year I've been looking for my next role and it feels like I don't fit in anyones box well enough to get the nod. Besides landing an exec position at a startup with funding, the big tech companies of the world seem like the only place to get good compensation for that expertise. From what I can tell, the big tech places don't seem to be interested in hiring people from startup-heavy backgrounds nearly as much as they used to be.

My approach is to try to reestablish some networks, and tossing resumes are interesting job listings. But the reality is that almost no one is getting hired for great roles without a warm intro to a hiring manager. Dusting off the old personal website never hurt either, it might at least let you get into the right kind of conversation after the warm intro.

It feels like I'd be in a great position to start a startup but with spouse and young kids its not the right time (especially now) to do that sort of thing.

I would start with a basic question: what do you want to do? Your current job let you explore a lot of topics and roles: which did you like best? Which do you see yourself pursuing for the next few years? Your post seems to imply that you’re looking for management roles but it’s unclear as whether it’s because that’s what you like to do or because that’s what you feel you should do to grow your career. Word of advice: it’s much easier to have a successful, fulfilling career doing what you love than eating shit every day doing something you don’t like (management, architecture, product, whatever) but that you think is good for your career.

Once you’ve figured out what you want to specialize in (and make no mistake, in a larger company you’ll have a narrower role than in a startup), write your resume around those skills. Market yourself (write a blog, try to speak at conferences, etc) as an expert in that topic. Identify what shortcomings you still have and work on them through side projects, reading books, taking an online course, etc. When you get to the interview stage and people point out your lack of expertise, explain (better yet, demonstrate through relevant examples) that working at a startup has taught you to pick up new skills quickly. Convince your future manager that it’s in her benefit to recruit someone who knows how to learn new skills rather than a one-trick poney, even if the poney is very good at its one trick.