Do you store the URLs I check?
No. The inspection runs, the result comes back to your browser, and nothing about the target URL is written to disk. The only thing stored is a hashed record of your own IP address with timestamps, used for the hourly rate limit and swept automatically after two hours.
Using The Inspection
Answer it for your own page
The inspection is free, needs no account, and takes a few seconds. It reports what is actually wrong with a page rather than what is generally true about pages.