A Ford Pinto in 1971 was $1,919 which is $15,000 in today's dollars.
A Yugo GV in 1984 was $3,990 which is $12,000 in today's dollars.
A Chevy Spark in 2013 was $12,995 which is over $17,000 in today's dollars.
Car companies can keep up with US regulations at about the $15,000 price point. Below that is losing money.
A Nissan Versa or Mitsubishi Mirage in 2024 cost about $16,000 which is right in line with the Ford, the Yugo, and the Chevy econo-rides of the past.
$3K-$4K wouldn't build you a crash regulation compliant unibody shell, much less a full car. If a major maker went balls out to build cheap and still be legal to sell in the US, they might be able to get the price down to $10K-$12K and still break even, but below that is a fantasy. Physics simply will not bend any further.