Why don't intelligent animals like primates use plastic bags?
Bags are not that useful if your are not human.

Most primates are too small to get any benefit from it, the bags are too large and would drag on the ground or get snagged on branches, even chimps and gorilla don't have enough ground clearance to not have this problem. Primates have been known to play with plastic bags, but they would not provide much benefit. A primate would need to carry the bag around for hours to fill it and wait before they could eat anything from it, all the while dragging it and hoping it did not tear. Upright humans are the only primate built to carry things long distance, a bag would not help other primates much.

Most primates are frugivores, carrying around extra fruit is not actually very helpful, most primates live near their food sources. Not to mention getting fruit in a tree into a bag is rather difficult task. Carrying around fruit is not very helpful because many primates have a different and much better way of storing ripe fruit in other primates AKA food sharing. Most primates will call other primates of the same species to ripe fruit since sharing in this way means they are more likely to have other primates share with them, thats far better food security than carrying around some extra fruit. It is also a lot easier to call many others to you than carry enough for everyone else.

The primates in your pictures are zoo primates which are both extremely bored and often very anti-social due to being raised in isolation. They might get some benefit, but they are not going to do well in the wild, and no zoo in its right mind is going to give its animals plastic bags.