When you use
https://fullscreen-for-googletasks.com/, it's a PWA, and there's a button on the top right, next to the ⚙️, that says something like "pin to desktop/start menu", you can use that to create a standalone app, so you don't need to open a browser / a new tab to use the tool.
When launching as a chrome extension, the underlying code is exactly the same as
https://fullscreen-for-googletasks.com, the only difference is the url being chrome-extension://ndbaejgcaecffnhlmdghchfehkflgfkj/index.html. The syncing behavior should be the same either way.
If you consistently find the syncing to be buggy, I currently have no answer. So far there's no other user reports of such issue over the last 5 years. Maybe try different browsers and see if they all have the same issue?
The chrome extension only works in chrome, but the website works in any browser.