Visit each site in your password manager and knock over a couple a day.
You'll never get anything that fully automates this because every site does the change differently. Even LastPass's automated password changer could only ever do a handful of sites fully automatically.
I've seen everything from full self service, to manual approval, to begging their privacy/legal department, to sites claiming allowing me to change my email address would represent an unacceptable security risk. I ask those sites that make up reasons like that to delete my personal information and move to someone who understands why email addresses are bad primary keys.
If you use your own domain, you'll never have to do this ever again, no matter how many times you change your email provider.
You need to start today, if you wait it's only a matter of time before Google does it for you by locking you out of your account for reasons they won't explain. Browse the GMail subreddit and observe the sea of hundreds of people locked out each week.
As for alternatives, I have iCloud+ with my Apple One subscription I already pay for. That lets you setup five custom domains, with up to three emails each[1].
I think even if you don't switch yet, it's a good idea to change your domain name while you still can because most services ask for confirmation on the old email and we've all heard the horror stories with Gmail locking you out of nowhere
If you're concerned about privacy I would evaluate fastmail.com
Any solutions to make the transition easier outside of manually updating email everywhere?
Also, what’s the best alternative to Gmail currently that will support using my own domain name?