It's somewhat more nuanced than that. The currently approved swabs have limited availability, and indeed there are others (unapproved) that would work. However it seems you can't use just any - they have to consist of synthetic fiber, can't have a wooden shaft, and can't contain calcium alginate. This rules out many of the swabs that a healthcare facility might have on hand.

(https://khn.org/news/as-coronavirus-testing-gears-up-special...)