How often are you not signed in to GitHub? You’re presenting this as a practical pain, but I can’t remember the last time I wasn’t signed in to GitHub.
I've also wanted to do code search on my phone, where I have no need to be logged in so I'm not.
Telling employees to make their own personal account to do company work seems like a bad idea.
In practice my employer would be happy for me to spend my time doing it anyway, because it's the right thing to do. But asking them to pay Github for the privilege is pushing it.
The use of a password manager makes re-logging-in effortless.