> Do American universities offer that as a button click online? ... transcripts to apply to grad school, or jobs in companies?
I believe yes, usually. You pay a small fee, like $25, for the service, of course. With some schools you may have to do it over the phone and/or mail a cheque.
The official transcript would be needed for further study like grad school, or for transferring credit if you transfer schools. I've personally never had a company ask to verify my degree or grades, though I suppose if I was a new grad from a prestigious school applying to a big-name firm, they might check. (I think there is a commercial service most schools use for this, which confirms basic things like graduation date and degree.)
The rules are because the school assumes that if it didn't come directly from the other school, that you could have altered it. Although I've also heard of schools providing sealed, tamper-evident, embossed envelopes to the students, and as long as you don't open that inner envelope it would be accepted as authentic by the other school. They refer to these as "official transcript" and the kind that are unsealed are called "unofficial transcript."
> print out a proof of enrollment yourself which includes a verification code that third parties can check.
This would be the "modern" way of doing such a thing, assuming you have at least say, 2000-era technology, but the way they have it set up is a custom that started probably 50+ years ago and to my knowledge no colleges have yet gotten together to establish a standard, so that they would be able to accept them from each other and verify them.