Test project loading and saving

We should perform end to end tests of project loading and saving. This seems like a complicated thing to pull off, but we should at least test the things we can.

- The fallback methods seem like it is well supported by playwright: https://playwright.dev/docs/input#upload-files
- The FSAA method seems to be impossible to support: https://github.com/microsoft/playwright/issues/18267 we can try to mock it, which is better than nothing.
- The Tauri method is not supported because it is proprietary. We could mock it, or maybe write tauri tests in the tauri project instead.

If we do set up at testing pipeline for tauri, we might also be able to E2E test the FSAA method there.