Any comments? Is it a good choice?

And why do they use that material, whatever it is? Why don't they use burlap?

Burlap will decompose when wet really fast. The rootpouches are made from a couple different materials and thicknesses that last various amounts of time, they photodegrade mostly. 

The landscape fabric in the video is spun bonded poly, most reusable shopping bags are made from it also but may not have as much UV resistance as the black landscape fabric.

How long does it generally take before you need the 3 gal pots?

When starting figs in my basement in the winter (January/February), they're usually ready to be repotted in June. I pot up from a 4x4x9 treepot directly to a 3 gallon.

For potting soil I keep my eyes open at the big box stores for broken bags of potting soil. As long as they are not pre fertilized I can get them for 50% off to totally free. I dump them in my wheel barrow and add perilite at about 20-25%. Then I bag it up for when I need it, whether it be for potting cuttings or starting/refreshing raised beds.
Once cuttings have rooted I plant them in 1 gallon pots I get free from my local nursery. Usually, here in Yuma, AZ I can transfer from 1 gallon pots to the ground in 2 months time. That's 4-6 weeks establishing in large plastic tubs and about 2 weeks of hardening off. This year I have started "groves", multiple plantings in a small area for some friends and am doing a 4 plant one for myself. I got the idea of "groves" from a Yuma area stand of old fig trees that have grown together beautifully. I posted pics of this grove earlier this year. 
sdpops Yuma,AZ zone 10

Have been unfortunate in the past with using soil from some big box stores. luckily found this site with its unlimited amount of resources! going to try much larger pot this year 10 Gallon, plastic.
Use this soil mix for my two mature trees:
5 - parts sifted Pine Bark Mulch 1/2" mesh
1 - part Peat Moss
1 - part Perlite
Plus - 1 cup Dolemite Limestone and 1 cup Espoma Plant-Tone OR Osmocote per 5 gallon of mix.

Looking for a very productive year!!