Is undergraduate research typically funded? Should it be?
All undergraduate research positions should be funded. Most currently aren't.
The children of families who can afford to have their kids not work aren't smarter than the children of families who can't. Spending $60 billion a year on government supported research in the US and nickel and dimeing the next generation of scientists so 75% of them have to give up science is a real stupid way to invest in the future.
As for mechanisms, the NSF and NIH already have them and they work. I write NIH student supplements every year.