The agenothree shooters are, in fact, described explicitly as not shooting fire. In Dragonflight, before Lessa brought forth the Oldtimers and their equipment, Mastersmith Fandarel couldn't get a design for a flamethrower that actually worked, so he considered using agenothree instead - as an acid to destroy the thread directly, not as a fuel source:
There are, it is true, fluids which burn and sear. We use an acid to etch design on daggers and ornamental metals. We of the Craft call it agenothree.
Also, it's specifically mentioned that when used on dragonback it would disperse, and even serve as a fertilizer for crops on the ground, as Fandarel tells Lord Vincet of Nerat:
"Little man, agenothree in diluted form is what you use to fertilize your plants in the spring. [...] It would be better if we could get the spray up high in the air. Then it would float down and dissipate harmlessly fertilizing very evenly, too."
And even some of the Oldtimers, with their fancy flamethrowers, could see the use of a thread-destroyer that could serve as a fertilizer:
D'ram, particularly, was very much interested in Fandarel's agenothree sprayer, considering it better than thrown-flame, since it would also act as a fertilizer.
(All quotes from Dragonflight. Sorry not to have specific references, but my book is out on loan, so I'm quoting from an unpaged ebook)