The point is that plastic takes many centuries to degrade, and we go through a lot of single-use plastic. For example, a disposable diaper takes >500 years to biodegrade.
It's nice to imagine all this plastic going into walled-off landfills that protect the rest of the earth and water from being contaminated, but in practice this is a myth. US localities are seeing an unsustainably growing amount of plastic contamination in local waterways and beaches, some of it visible, some of it not.
Incidentally, this doesn't mean you need to use a single-use paper bag instead of a single-use plastic bag. Instead, use a durable bag made out of any material you like. The phrase is "reduce -> reuse -> recycle". Recycling was never meant to be the foremost part of our sustainability efforts.