How to get an engineering position with CV showing product experience?

So COVID-19 sucks, and it's time to start looking for a new position. I started at a startup 9 years ago as an engineer and I was either the best engineer or one of the best for the first 5-6 years, back when I was doing engineering work full-time. Designed and implemented large systems from scratch, that are doing well in production nowadays: great uptime, still meeting functional requirements, and flexible enough to support new features easily.

I don't regret this personally, but it might have been wrong for my engineering career: I then had several roles that aren't hands-on engineering. I was a technical product manager for a cool product that I helped create and launch. I had a team of 4-5 engineers, but I personally wrote a major component myself because it was really sensitive and had to be implemented using tech that the team wasn't familiar with. They were over time introduced to it, and today they maintain it successfully, even if I sometimes help with code reviews - but for the initial launch it was much faster for me to write the first version. After that I moved into a people management role managing several product managers and some architects, which taught me a lot and I did it really well.

So now my CV shows for the past 3+ years that I have management and product experience, and I get rejection notices from Greenhouse almost immediately for positions that I apply for. I don't want to lie and remove these product roles. I also think my varied experience makes me a better engineer: one who understands the customers, the business, the sales process, and the complexities of team work, better.

Other than reaching out to anyone I know on LinkedIn and hope to skip the initial HR barrier, what can I do to get my next job?

I'm in the US, in a top-five city.