Point-in-case, I saw 2-3 "DevOps" engineers do it in my last position... over a screen share..! When you'd bring it up to people they would just roll their eyes and call you paranoid. =(
Visiting a webpage and copy-pasting a string off of it is not a very good practice for security because you're adding on a lot of parties to trust with that secret!
Effectively you want to minimize ANY place that your secret exist in plaintext, and trusting a webpage with this is just not a good idea.
I imagine owning random.org, and not being very mean but a little clever. I know how many people come here for a quick clip; more importantly I know you come here. I rotate the same blob. I know all the pieces to brute force your infrastructure. Maybe you’ll use the wrong setting and something will be public that shouldn’t be. Hello.
And yes, our recommended way is to have the secret management automatically generate the password. That way, not even your workstation touches it.
It all comes down to chain of trust. When it comes down to a root password, or any secret that's business-critical you want to minimize ANY sort of risk and that's just the right way to do business.
When I point my scraper at random.org I can see it talks to "ocsp.digicert.com", "ajax.googleapis.com", "ssl.google-analytics.com", and obv. "random.org" (wow that's actually pretty good =P)... those are now three separate entities that now need trust because they all have the opportunity to see what was rendered on that page in plaintext, they have the opportunity to see what you selected, etc.
Then add to that, any browser plugin, the browser itself, etc etc. Then the "in plaintext over screenshare" issue - and you've got a lot of points where something, or someone could MiTM a plaintext password if they wanted/needed.
Generating a random password/secret by visiting a public site on the internet is stupid/silly with regards to actual security, and opens yourself to attack vectors _for no real reason_. There are a TON of VERY QUICK/EASY ways to generate a very secure string for secret management that don't involve trusting a ton of third parties =|
In a "security culture conscious" SF tech company there should be no place for laziness/lack of care like that. IMO - dumb compromises like that are how you get caught with your pants down leaking a ton of PII.