I am aware that to sell food items supermarkets and such require licences and must pass health inspections. But what about the products themselves?
I read this which mentions some low risk products can be sold before being tested. Does this imply all other foods require a lab test and government approval prior to sale? I believe the FDA is the department that keeps dodgy stuff out of US markets.
No, products that comply with FDA guidelines are not tested before being offered to the public for sale.
It is impossible - literally* - to test every batch of ice cream, spinach, frozen hamburger, etc. that is shipped out to supermarkets for sale. The FDA relies on the good intentions of the manufacturer to follow its guidelines and results of evaluations. If they do not, they are held responsible for damages inflicted on the public, as well as punitive damages and fines.
In the example of ice cream, the FDA has guidelines for the facilities that produce the product based on years of studies and experience. They include cleaning recommendations, temperatures at certain stages, the amount of time the product must spend at certain temperatures, etc. The facilities are inspected routinely, and when a product ends up contaminated, the FDA does a new inspection and reviews the previous inspections for evidence that previous recommendations were ignored, or might have prevented what occurred. In other words, it's a continuous process that relies on the manufacturer to keep in compliance with FDA guidelines. Facilities that don't comply face recalls, fines, and ultimately closure.
The Blue Bell Ice Cream Listeria outbreak is a good case study in the process. The process can be examined here and by following the links within.
*The costs would be prohibitive to test for every possible pathogen and contaminant.