This is interesting, they were using bast fibers from oak trees, or fibers that sit between the bark and wood of the tree. They are soft fibers that can be collected and weaved into cloth.
Wikipedia has an article (1) and pictures of modern bast fiber weaving in Norway.
Those same bast fibers from western red cedar trees were used by native people in the US Pacific Northwest area for clothing. Hats and cloaks were common. The clothes have a texture like soft paper and are water repellent.
Polynesians have long made "tapa" or "kapa" cloth from mulberry bark. It involved a great deal of hammering using a custom stone hammer with a flat face a couple inches wide and maybe 8 inches long, with a grooved surface. I expect the bark was laid over a flattened log.
The idea that 7000 BCE is "shortly after the last ice age" is funny. We could as well say the present day is shortly after Julius Caesar was assassinated in front of the Roman Senate.
Something that really is coincident with the end of the last ice age (technically, the end of the 1200-year Younger Dryas cold spell brought on by a bolide strike on the North American ice sheet) is Gobekli Tepe. That was, thus, built some 2000 years before this fabric was made. The 200m sea level rise that started ~20kya was ending about then, with the final bits of what had been the Persian Gulf river valley filling in.
One often comes across the term "valkala" in Indian scriptures, which means clothes made of tree bark, usually in the context of asceticism. E.g., see here: https://sanskritdictionary.org/valkala
Totally off-topic observation: I find it curious that OP changed headline from "8000" to "8k". My scanning eyes converted 8k into a measurement of money/data, instead of one of time.
Because long sheets are available so you don't have to invent weaving? Or because the bast is long staple and can be spun? Bark "tapa" cloth is beaten into sheet form in Oceania and like places.
I'd argue that having some light clothing to wick away moisture is better than wearing your birthday suit. Additionally having a bit of protection for certain parts is a good idea when you're running through brush and doing things that are likely to do abrasive things to your skin.
It's less common, but there's also a lyocell process for bamboo. (I have sheets and towels that are bamboo lyocell, and they feel amazing) It's also likely only possible using modern technology, but it is far more environmentally friendly than bamboo rayon (aka viscose).
Technically, there's no taxonomic group for trees. Palms are grasses too but are considered trees. Though I suppose bamboo doesn't fit into the concept of what a tree is depending on how one looks at it.
Both palms and grasses are monocots (only one seed leaf as opposed to two and parallel veins as opposed to the type of veins you might see on an oak leaf).
Palms are not members of the grass family (Poaceae) though.
Wikipedia has an article (1) and pictures of modern bast fiber weaving in Norway.
1- https://en.wikipedia.org/wiki/Bast_fibre