It turns out you can 
The answer is Actual Budget, which is free and open source and built the same way as YNAB!
After @Drewster and @jmanko16 recommended it I’ve looked into it again. When I looked into it in 2022 it just seemed too intimidating to implement (for example I think at that time there were only GitHub files to download which is very scary for non-tech people!). The onboarding has been much improved since then. Now you can just download the app straight to your Mac (no GitHub required!), and they’ve set up a hosting option so that you don’t need to do any of that yourself if you don’t know how it works.
Because it’s now easier to get started, I decided to give it a go in my testing, and for me I clicked immediately because the app is basically YNAB without the frills. I haven’t really had to learn how to do anything because I’m familiar with the mechanics already. I haven’t decided how I feel about my budget being on Mac when I usually manage it on my iPad, but it turns out I like the idea of it all being locally stored on my devices. I’m considering if I might want to set up the server option so I can use it in iOS but I’m also thinking that being Mac only is much more sustainable long-term and maybe I will just leave it like that. I’ll see how it bothers me over the coming month.
Here’s a couple of follow up notes on the other app I’d mentioned that I tested this week:
Lunch Money: Good-looking software with a generous pricing scheme, but the manual was not very good and it didn’t seem designed for zero-based budgeting. I couldn’t figure out how to implement it well and gave up. It’s browser only, I’m not sure how I felt about that, I think on the whole I was probably ok with it but I’m not sure! In the end it didn’t matter since it didn’t seem to have been designed for zero-based budgets anyway.