As soon as you see a sprout of something you don't want growing, pull it while it's easy, tiny weak roots, and only takes a second. Letting a bunch of weeds grow until you have to "go weed them" is no fun, a lot more work. The roots can also become enmeshed with the weed barrier very quickly, making them more difficult to pull. If left alone, they will go down through it, getting quite a foothold and shredding the weed fabric. If weeds drop seeds, you can expect an even bigger crop of them the next time.

As you've experienced, all it takes is a small amount of grit to settle atop a weed barrier to make the perfect place for weeds to grow. Weed seeds come from the wind, and are dropped by birds, so the battle is never over. That barrier stuff really doesn't work any better than mulch alone, and will prevent a significant portion of water from going through it, slowly killing plants. As the mulch decomposes above it, the resulting nutrients are much more available to weed roots above than the desirable plants' roots below.

The roots of the desirable plants may become enmeshed with the weed fabric also. Once the fabric starts to decompose, it will get ragged and start poking through the mulch, it's really a nightmare and very difficult to remove at that point, and ends up coming away with sheets of roots. Every time I've dug this stuff up, the soil that has been under weed barrier for a long time is very poor, lacking organic matter and tilth, hydrophobic, draining poorly.

The dead plants I've excavated had terrible root systems, shallow and weak.