Some crafting classes are more dependent on others for materials than others are, and (as you've already seen) Weaver depends on a lot of Leatherworker items. It's true the other way around as well. Leatherworker needs a fair amount of thread and cloth from Weaver. So it tends to be easier to level those two up together. (Each needs occasional items from other classes as well, but not as often as from each other, so that makes it somewhat easier to either buy those occasional items, or just avoid the crafts that require the more expensive ones.)

Even if you don't feel you have time for dealing with all eight crafting classes at once, would you consider expanding from two to three of them at a time? If so, I'd suggest the following:
  • 1st) Unlock Leatherworker, Alchemist and Culinarian, and level those three from 1 to 15. (Pause on leveling Weaver and Goldsmith during this step, except for whatever Weaver items you need to supply your Leatherworking.) Alchemist and Culinarian are relatively self-sufficient in terms of materials from other classes, and Leatherworker mainly needs Weaver items, which you can already make.

    Once you get these three up to level 15 (along with Weaver and Goldsmith which you already have), you'll have access to five of the best level 15 cross-class skills. So you won't be too gimped in your ability to produce High Quality items while you're leveling stuff the rest of the way. (You still won't have quite the versatility of someone leveling all eight classes, but you'll gain a lot of it.)
  • 2nd) Once you've unlocked those cross-class skills, abandon ALC and CUL, and focus on bringing Leatherworker up to around the same level as Weaver.
  • 3rd) Continue leveling Goldsmith, Weaver, and Leatherworker (trying to keep WVR and LTW near enough the same level to supply each other with mats). You'll still need occasional metal or wood materials, but not that many, so you can either buy those or just avoid the recipes that need them, depending on how expensive they are.