It is useful to level weaver at the same time, mostly to save gil by making your own gear on your way to 50.To make end-game armor you need to know that a level50 weaver isnt enough. To make ilvl90 gear HQ, you need the ilvl70 crafting tool and alot of materia melded into your gear. This is a hell of a grind tbh. You also need crossclass skills from other crafts, so leveling weaver only isnt enough.

If creating end-game gear for your healer is your only incentive to level weaver, you will probably be better off by getting gear from myth tomes or buy crafted 90 gear from the AH. At least on my server you can get HQ gear on the AH pretty cheap now. (around 100-150k + approx the same in materia for melds)