I reckon it will grow through, and you’ll never get it out once they grow through it! Something like old woollen blankets might work. They’d be more closely woven and not let weeds through and would break down eventually. I wish I could they were easy to find, but sadly they are not. Sheep fleece (which is worthless and generally easily available) might work though. It would be harder to handle, and you will find bits of it around the place. I’ve used it as a mulch quite successfully in smaller quantities (I’m sure I’ve seen a commercially available wool mat designed for this. I recall the words ‘not cheap’ being used in conjunction, though.
I have a love-relationship with landscape fabric. It is so beautifully weed-suppressing and saves me SO much work. But it’s plastic. If you are really struggling with weeds, I’d be tempted to go with cardboard (two layers, perhaps) and then as much compost as you can manage and then plant through the landscape fabric on top of that. If you make planting holes in it with a blow torch and use it to heat the cut edges you don’t get bits everywhere, as it seals them and stops it fraying. Don’t mulch on top, and you can take it off at the end of the season, and hopefully not need it next year. I’m aiming to get to a point where I don’t need to buy more of it, but planning to use what I have until it falls apart.