How can I stop kitchen sink caulking from peeling off after a few weeks?
If the plastic sheet you put over the curing caulk formed a vapor seal, the caulk did not cure properly which could make it not adhere and peel.

Curing caulk should be left open to the air to off gas at its natural rate determined primarily by temperature.  As long as the sink is at room temperature this should happen in less than 24 hours.  Do not disturb or splash it within the first 30 minutes.

As long as the counter top is an ordinary material&mdash;stone, wood laminate, or composite&mdash;it just has to be clean and dry before caulking.

I certainly would expect a professional sink installer to caulk.  Perhaps they refused because one of the counter or sink materials is known not to work?  Oily woods like Rosewood, Ebony, Cumaru, Ipe, etc. are well known to be difficult to glue together.  A solvent can be used just before caulking to improve performance.  See this.