Actually you don't want them to last forever. You want them to decompose after time to grow them into the next size basket.
You will find that sooner or later you will need to remove the old material and repot with fresh. Moving an old basket into a new one without changing the mix will work for a while but as the original old roots in the center of the rootball age and start to die off, this area will become the seat of future trouble in my experience. The free root run of the basket will/does greatly improve the vigour of the plant but to keep it going long term (10 years +?), I find that it helps if you can get to that middle of the rootball area somehow and blast the old dead roots and rubbish with water.If you can manage this with the wooden basket without removing the plant then no problem. I also find it helpful to stuff this old center area with bits of polystyrene and surrounding that with your normal mix.
With the plastic baskets, you get the benefit of all the air around the roots and the advatage of being able to lift out the entire root ball and work on it before you move it up to the next.
The plastic ones hold up too well, and when you try to move the plant out of a plastic net basket the roots can get torn up.
I agree so when I use the mesh pots now I line the inside with weed-mat fabric to avoid that. Not as open as a basket but better than a big pot.