If the doctor was not using a gmail account, the UI probably recomended to share it "with anyone that has the link" that is like public but protected by oscurity.
Most people don't realize that it is 99% like posting it on Facebook.
The URL may be logged by the ISP, university/company or whoever is providing the connection, antivirus, windows recall, browser history... But IIRC passwords are send using cryptography if the site uses https.
The URL is sent using http inside the https connection, so it's encrypted too. Your ISP shouldn't be able to see it unless they have root certs on your device.
It's not impossible that someone sees the URL, and if it's stored in an email the backups or local clients could store it plaintext, but I don't see how the URL is any more likely than the password to be sniffed during the connection itself
Its nothing like posting it on Facebook. Google doesn't have a recommendation engine that tries to share the image with random people, or a search that image will appear in, or a set of subscribed people who will be notified.
With LLMs getting good, I've noticed people are more akin to fixing and new adding features to open-source projects that will never see the light of day.
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