And even if not everyone does, operating entire flights at business class+ fares is probably not viable. Especially given that lie-flat seating (e.g. Polaris) with at least decent food is pretty comfortable for the people who are willing to pay a premium.
JSX operates an entire airline as business-only. They use CRJ-900s, so not long-range, but there's no TSA (they are technically charter flights, you board at an FBO).
You could feasibly run major transatlantic routes once or twice a day like that.
JSX operates based on a loophole in the part 135 rules, but that only allows 30 seats. A CRJ doesn't have the range for (nonstop) transatlantic, bigger planes would be impractical, and smaller ones with the range won't hold 30 people.
BA has, as I understand it, gone back and forth on London to New York business only flights. But that's not designing a whole new aircraft for the purpose.
A CRJ-900 is a very, very common regional jet. They just changed the seats.
I'm not making a comment about the original topic, just the comment that operating an entire airline as business wasn't viable. Not on every route, no, but for the right route? Yeah, there are enough people who fly certain routes and will pay for a better seat that you could fill a plane with them.
If you have demand for 10,000 passengers to be transported on a route daily, and a plane carries 200 people, you would need 50 trips. If the round trip is 4 hours, you could do 6 trips per plane and you'd need 9 planes to service the route.
If the round trip time is 8 hours, you could only do 3 trips per day and you'd need 17 planes.
Pretty simple - we care more about air travel being cheaper and safer than we care about it being faster.