In the U.S. they give them away for free in most places. Reusable bags are starting to take off, but there’s no incentive to using them besides morality and having a preference for when you find a really good reusable bag. (We’ve got some with a bunch of internal pockets, and they’re the bomb. We also have several that are insulated for cold stuff, and they’re great, too.)

Didn’t we stop using brown paper bags to save the trees?

Although the material in itself can be renewable if coming from managed forests, the work needed to harvest wood transform it to pulp and then to paper and the increased weight of the final product to transport is also extremely energy intensive, several journalistic pieces and studies attribute a much higher environmental impact to paper bags compared to single use plastic bags. The worst offender are cotton tote bags which would need to be used several thousands times to offset their production.

https://phys.org/news/2020-05-plastic-paper-cotton-bag.html

Probably not, honestly. Paper is currently sustainably produced (most of the time), and paper bags are often made of recycled paper.
But plastic is very cheap to manufacture, and the environmental angle (plus plastic bags used to be thicker, so they were more durable than paper bags) is likely what sold consumers on plastic while retailers cut their costs.

Regarding environmental concerns - I did a smallish dive on it, and it looks like plastic is more environmentally friendly up to the point of use. It’s more efficient to make (less industrial processes to convert raw materials), more efficient to transport (less weight per bag), but isn’t really recyclable or compostable, and isn’t quite as reusable. National Geographic Article over it.

And, with regard to what you are saying, it looks like the plastic he’s using is called PBAT. Like all plastics, it can be combined with other stuff for various properties, but it looks like it primarily can only be composted in industrial composters. It may also be prone to not composting fully, releasing microplastics. It may also release toxins into the compost (toxic to the single celled organisms that help with bioavailability, that is), lowering the quality of the composted soil. It also has a sort of low melting point, which means it can leech chemicals, moreso if it’s touching liquid things, or especially, warm things - which makes its use for food containers pretty concerning.