Well, actually, they will let you - but you have to do it on the Government's terms.
If you wanted to import a car, this is what you'd do -
Purchase the car and have it shipped to the port closest to the nearest Registered Importer.
When the car landed, EPA, DOT, and customs bonds would be issued for the car. If the car was not made compliant, the bonds would be revoked and the car would have to be deported or crushed.
RI's are shops that are licensed by the Government to modify and inspect non-conforming cars and turn them into federally-compliant cars.
Now - the RI's can charge you as much as they want - the Government doesn't prevent them from price fixing because they want to keep personal imports out. For a brand new car, the RI's job might cost $20,000 or more. For an older car, it'd be less - as the engineering and modification required will be less.
A 1979 Citroën GSA or a 1980 AlfaSud might only cost a couple thousand dollars to import.
A New VW Lupo GTI or a five-year-old Nissan Pulsar SSS is big bucks territory due to the complexity and the stringency of the requirements.
The RI would do their thing and demonstrate compliance to the Government, at which time they'd let you have the title and the bonds on the car would be lifted. Then you would be free to go register it.
The things that really put an end to personal imports were the repealing of the personal import exemption (one person was free to bring in one car and that's it, with no regulation, one time) and the adaption of the On-board-diagnostics laws. OBD-II and OBD-III are very expensive systems to engineer onto a car not designed for them.
Even before OBD-II, it wasn't cheap to import a car. A company called CXA imported Citroëns to the USA in the late eighties and early 90's - and when the Citroën XM came along, they had to add around $18-22,000 to the price of the car to cover the cost of certifying it - meaning that what was really a $30,000 Citroën became a $58,000 Citroën - and the market appeal was much less. They sold fewer than 25 of them.
This brings me to my last point - manufacturers and their north American representatives are unlikely to be helpful. CXA had to fight Automobiles Citroën in France and Peugeot Motors of America every step of the way and their efforts were done in the face of objections from the factory. The manufacturers don't want to be involved with what could be costly long-term parts and liability questions.
People have gotten around this through shady methods, but this is the legal way.