This is an interesting issue, so I just fired up your application too and I see the same:
You'll notice that in any of the Chromium-based browsers no POST requests are generated, only a few GET requests. You can see this in both the Cypress UI runner (left side) logs and in Dev Tools > Network, versus manually doing this in normal Chrome.
So I wonder if one of the default Chrome arguments passed with Cypress Chrome is blocking your application from making these POST requests!
See the docs here
https://docs.cypress.io/api/plugins/browser-launch-api#See-all-Chrome-browser-switches, there are at least 50 arguments / switches set in default Cypress. It's not easy to go through all those arguments 1 by 1 and disable / enable. So I agree with
@fresh-doctor-14925 that this deserves a Cypress github issue. And I'm interested to know the outcome of this one too 🤔