There are some important technical issues you should check for, at least weekly.
You can’t “optimize” a website that Google can’t read. Thankfully (hopefully) your web developer gave you a site that’s easy for Google to read. However, developers’ goals are to make it “work” for people- but that can mean it doesn’t “work” for Google. A weekly check can help see if your developers have done anything that might prevent Google from reading or accessing your site.
At the same time, Google’s rules sometimes change. Sometimes this can benefit you. Othertimes this can hurt you. We’re playing a game where the rules can be changed on us- without telling us. A weekly check can help you get ahead of these changes before they hurt you.
Of course, sometimes people make mistakes too. Did someone make a change without understanding the SEO implications? Did they forget to tell you (again)? A weekly technical checkup can help catch human errors, too.
When you first start doing the weekly SEO checkup, you might find a lot of things that need to be fixed. This might take a lot of time. It might even be expensive. It’s worth it: Google can’t rank a site it can’t read. Give yourself a good foundation and you’ll have an easier time with the other things, later.
After you work through the initial bugs I think you’ll find that this weekly checkup will be very quick. You’ve cleaned up the biggest errors. Now there’s just a couple fine details to fix. Maybe everything’s looking great and you’ve just taken 5 minutes for peace of mind.
When you combine this weekly technical audit with the weekly Google Analytics checkup, I think you’ll catch most errors that could come up.