How often does it happen? If you've had all the standard prenatal tests, and come up with no warning signs, then the chances of having an IMMEDIATE problem after birth are pretty low. I don't know that anyone could give you a statistic, though. Most congenital problems will either show up in testing or they won't present until after a day (or a week or a month, etc.). Your "most common" congenital defects are probably neural tube defects, which are tested extensively for during pregnancy. After that are probably trisomies or other chromosomal abnormalities, which are also tested for during pregnancy. The biggest risk (if you've had testing) is having an undiagnosed completely unexpected congenital defect like a heart defect, and the risk of heart defect is (I believe) about 1% if you do not have a family history. Even so, many heart defects will be found at the 20 week u/s, since that is in fact one of the things they are looking for.
Most commonly, the help that a baby would need would have no underlying "syndrome" to cause it... occasionally a baby will just need help clearing his/her lungs, and a midwife would be completely prepared for that possibility.
But, please keep in mind that if you have a midwife attending you, she should be prepared to handle an emergent situation, and 2 minutes from an emergency room is closer than a lot of people who HB, and even closer than some mamas in the hospital.
ETA - in the case of the mama who started that thread, the problems her LO had are exceedingly rare. I was one of the mamas who responded to her - my LO's problems were diagnosed at the 20 week u/s, and that's why I chose to birth at the hospital, but so long as the 20 week u/s shows nothing this time, I'll be birthing at home.