Sisal comes in tighter grade weaves.
ALL carpet however, has the ability to 'filter through.' That's just the nature of how it's made.
Carpet is either woven, and then backed, or it is a tapestry type medium, but still has a backing as the stabilizer.
If this was an issue that concerned me, I would choose the tightest weave I could, and jute is not it.
You are correct-wool is very tight.
But there are many synthetics made today that are super, super excellent at stopping sand, minute debris, etc.
Try looking up Fabrica- particularly the Chez line (pronounced Shea) and Masland rug lines. These are needle points, have an excellent base backing, and have a cost to match the high quality.
I know this from years of creating high end area rugs-and these are the only two styles I will work in. They are certainly worth the money, so even though you love natural, wait until your hands brush across Fabrica Chez...you'll swear it's animal fur...but it is synthetic. And Fabrica Chez, years later will have no issues of sweeping underneath-it is the most tightly woven carpet I have ever seen-hence my use for it in area rugs.
If however, your'e allergic, I suggest a very tightly woven sisal, or wool. It will still have debris issues, but at a much lesser rate.
Again, all carpet is going to have an issue as you describe, simply by it's manufacturing nature.
On another note, (I don't know your particular taste or need) commercial is easy to clean, due to the short strand.
Good luck to you