Well right, and coating them with plastic means that they leave plastic residue behind if they break down in an uncontrolled environment, and increases the cost and complexity of recycling:

If the paper has a plastic or aluminum coating, it can be recycled, but it is much more expensive and complicated.

Some plastic coatings can be separated from paper during the recycling process. Still, it is often cheaper and easier to use virgin materials to create new products than recycling paper coated with plastic.

Paper coated with plastic isn’t suitable for composting, and most times, such products are incinerated for heat or landfilled rather than recycled.

https://www.almostzerowaste.com/non-recyclable-paper/

Yes they already exist. They are not really better than pure plastic, they're kind of a form of greenwashing because they appear to be environmentally friendly.