A visa allows you entry, an ESTA doesn't allow you entry but just says you're allowed to travel to the border to see if you'll be allowed entry or not.
It's more for the airline than for you. If you have an ESTA and get refused entry, the airline isn't fined for bringing someone unauthorized to the border.
> A visa allows you entry
That's incorrect, holding a B1/B2 US visa only allows you to arrive at the port of entry, the actual permission to enter the country is always left to the border officer interviewing you
> A visa allows you entry
Nope
It's the same deal as you mention above
https://travel.state.gov/content/travel/en/us-visas/tourism-...
It states regarding a visa: "A visa allows a foreign citizen to travel to a U.S. port-of-entry (generally an airport) and request permission to enter the United States. A visa does not guarantee entry into the United States.
The Department of Homeland Security (DHS), U.S. Customs and Border Protection (CBP) officials at the port-of-entry have authority to permit or deny admission to the United States. "