Career path for Quality & Automation Engineering managers

Your best bet would be to talk to Automation / QA folks directly. Coffee chats (live meetings) are especially well-suited here since you get info that people typically wouldn't put in writing.

Put this in a message in your Taro Networking profile! https://www.jointaro.com/networking/

You can do a search on LinkedIn for terms like "Automation Test Lead" (e.g. I worked with Jing at Meta!)

My answer as someone who hasn't done this job function: it's similar to SWE roles except more of a bias to management. My sense is that there's less room on the IC ladder in QA/automation.