This is not as easy you make it sound. First, when entering the US, the immigration officials verify whether student status is valid using the SEVIS database. Second, all changes to student status must be reported to the SEVIS database by the school. If the school finds out that a student is doing unauthorized work, or if they find that the student's academic progress is unsatisfactory (likely if they are working full-time) they are required to terminate student status and report this to SEVIS.

The generic issue of using forged credentials to get an unauthorized job is applicable to any visa. And if I wanted to overstay my visa, it would be dumb of me to choose a visa that was as carefully tracked and monitored as as the F1 visa.