The "piece of paper" was the stub of your I-94 that is clipped in your passport.
You must surrender it when leaving, and... immigration knows.
They just enter your data in their computers.
The ban is not for life... the ban is 10 years, to apply for a new visa.
New visa could be denied. Suggest an immigration lawyer if needed.
Wow... 2+ years overstay is not "a few weeks"...
xxx
Happy contrails