Can't these websites store all these generations inside a database and then use them later when they detect a balance?
Yes. Avoid!
The fact that this page the OP has made has no way of downloading this generator as a standalone HTML file, should be seen as a very large red flag. He even put 'github' in the URL as an attempt to have this site viewed as some sort of technical-savvy cryptographic suite.
"Pure Client Execution." - lol right.

No... GitHub put GitHub in the URL - that's how it deploys your HTML app as a GitHub Page:
https://github.com/nwowner/airgap-machine-entropy-bip39/blob/main/index.html is OP's source. As of now, the only other file in that repo is the readme.
See also thebutterzone.github.io =
https://github.com/thebutterzone/thebutterzone.github.io (I didn't make mine a subdirectory page like OP's app, OP's main GitHub page deployment is nmowner.github.io =
https://github.com/nwowner/nwowner.github.io)
What I will say though, is the entropy generation from mouse movement seems way too quick. Just a few seconds vs around 25 seconds with bitaddress.org
It doesn't matter that the site doesn't store data (which can be confirmed by reading the HTML source & also looking at your browser's option to clear cookies & site data) if the entropy is extremely low?