Not to mention the bags are usually made from significant amounts of recycled paper. So you're already getting a second use out of the material.

I feel like in the 80's when there was a big "paper kills trees" moral panic the plastic companies took advantage of that to market plastic, and paper companies started recycling their products and developing fast-growing pulpwood species to plantation farm sustainably. And now we could absolutely go back to paper with fewer ill effects. It's probably better for the planet because even if your paper bags end up in the Great Pacific Garbage Patch something will decompose them into their organic constituents very rapidly and they won't agglomerate into rocks on beaches somewhere.