I had this same problem, and it turned out to be that the hospital was stealing every bit of cloth and thread in the entire fortress.
Even then, the clothier industry is very finicky. To make a nice cloth shirt, you need a farmer to grow the crop, and then another one to thresh it, then a weaver to weave it, then finally a clothier to make the shirt. If you want to dye it, you need a miller and dyer, too. At any moment, for the most trivial of reasons (say, the cloth that the dyer wants to dye is currently being transported to some stockpile), any one of these workers will cancel their job and hold up the entire production line.
Compare this to the creation of a door, or traction table, or iron short sword; it's an incredible headache. The only way I can deal with the clothing industry is to use dfhack's workflow to force these guys to make clothing no matter what, and turn off all job cancellation notices. What a lot of other players do is make everyone join the military once, then have then wear mail shirts and leggings in place of clothing. It's not pretty either way.
Well, you can make it way better though. I don't use workflow (in fact I pretty much play vanilla + various fixes to assorted bugs and therapist, that's pretty much it).
I.e. don't allow bins to be used on pigtails/ropereeds/thread/cloth stockpiles, unless you do have a huge amount of stuff which needs to be stockpiled. Most of the issues related to this industry is bins-related, and you are indeed
not required to store everything into bins. Everything goes much more swiftly if you just don't make haulers carry around bins all the times causing cancellation spam.
I just let my stocks build up a bit before I start producing the next product in the pipeline, and even then I rarely put jobs on repeat. It takes quite a bit of experience and understanding your production-throughput in all points of the pipeline to finetune an industry this complicated to the point you can just have jobs on repeat without risking for things to go wrong. Well, that's in my experience, at least.
Given these 2 very simple tips, you can pretty much build things up (adding more plots, processors, weavers, dyers, clothiers, haulers) as you see fit.
Alternatively, if all you care is keeping your dwarves from going around the fort while naked, you can just ditch the industry as a whole. Caravans will bring plenty of cloth/leather for you to shortcut the most tedious parts of cloths making.
I had this same problem, and it turned out to be that the hospital was stealing every bit of cloth and thread in the entire fortress.
Even then, the clothier industry is very finicky. To make a nice cloth shirt, you need a farmer to grow the crop, and then another one to thresh it, then a weaver to weave it, then finally a clothier to make the shirt. If you want to dye it, you need a miller and dyer, too. At any moment, for the most trivial of reasons (say, the cloth that the dyer wants to dye is currently being transported to some stockpile), any one of these workers will cancel their job and hold up the entire production line.
Compare this to the creation of a door, or traction table, or iron short sword; it's an incredible headache. The only way I can deal with the clothing industry is to use dfhack's workflow to force these guys to make clothing no matter what, and turn off all job cancellation notices. What a lot of other players do is make everyone join the military once, then have then wear mail shirts and leggings in place of clothing. It's not pretty either way.
I'd say traction benches are more of an headache for me than clothings, I uses managers to queue things up and ignore job cancellation. Traction benches need chain or rope, tables and mechanisms, which means I need clothing industrty or metalsmithing, a mechanics shop/forge for mechanism, and mason/carpenter/metalsmith for table and finally mechanics shop to put it all together. Clothing industry's relatively linear and straightforward in comparsion to me, a lot easier to arrange stockpile linkage for sure from farm > threasher > loom > clothier shop ( Dyeing don't really do much for dressing up and ropes with image on it are already absurdly valuable without it ).
Of course, it depends on fortress design and how stockpiles are set up as well as personal preference, metalworking industry in general is an headache for me and I usually uses leftover or junk cloth ropes to make traction benches since it only takes material value from the table
Though maybe I'm just used to it, really, it does depends on overseer's priority and designs.
I don't usually even bother producing ropes/chains for traction benches. I just buy whatever junk-rope I can from caravans and use those in benches and as restrains. Now, if I want to build an awesome well, I would actually take the time to have a masterful steel chain, encrusted with gems or something.