So when I run the Org check while its running the page is stuck on it and I cannot do anything else while its running?

is right about just waiting until the end of the process.

About support the best approach is as

suggested to reach the org check group on the trailblazer community or to log a ticket on GitHub.

Please do not send email directly to me this is the best way to not have answer fast.

Also as all Salesforce labs like org check the Salesforce support does not cover it at all. No use to create a ticket to them.