This is a real punishment, because if the alien agrees to leave the U.S. and apply for an Immigrant Visa at the American Consulate in the home county, the consul would probably hold back the visa for many years on account of the "3-10 year bar" and because of lengthy adjudication of the required waivers ("pardons").
An experienced Immigration lawyer would be able to weigh the chances and the risks in taking the option of leaving the U.S. and applying for an Immigrant Visa at the U.S. Consulate. Sometimes the best advice would be to just wait in the U.S. for another opportunity.
If you receive a Notice To Appear (NTA) in Immigration Court, the actual facts of your individual case are very important. (To read more about Immigration Court - click here). Your Immigration lawyer may be able to find legal arguments to enable you to stay in the U.S.
For holders of K-1 Visas it is very important to get good, comprehensive legal advice.