You will never, ever, completely eliminate those unwanted plants known as "weeds" because some of them send out seeds while others are spread around by birds and animals. If your expectation of a mulch is that it will eliminate "weeds" you will be greatly disappointed. Mulches often provide a very good place for plant seeds to germinate and grow. A mulch can help limit unwanted plant growth, or any plant growth for that matter, but will not be the only thing necessary to keep all unwanted plants from germinating and growing.
Along with a mulch you probably will need to periodically take a cultivator and scruff that mulch so the unwanted plats roots are disturbed so they won't grow.

Bark pieces are usually too big to make an effective weed barrier, or emulate any of the other desirable qualities of mulch/mulching. Finely ground hardwood, or a thick layer of leaves would give you much better results.

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.

The size of a particular mulch material is of less importance then the thickness of that material. I have used fairly large sized bark as mulch with very good results as well as very small bits of bark with equally good results, as long as the mulch layer was 3 to 4 inches thick. My personal choice for mulch material would be shredded leaves.