Yeah if only they did, but they didn't! Ferelden has one eluvian in the Dragon Bone Wastes, one in Soldier's Peak, and one in Skyhold. Orlais has one in Halamshiral, and that's it.
You are incorrect and vastly underestimate the scope of the Eluvian network.
We travel through several Eluvians during Trespasser that lead to locations in Ferelden and Orlais, accompanied by notes that suggest these Eluvians were scavenged from Elven ruins and sold to nobles as luxury items. And those were just in part of the network we have access to.
Solas refers to Briala and the Qunari having gained access to small sections of the Crossroads, meaning the full network has not been fully explored. This is backed up by the Qunari being shown mapping the network, as well as gathering Eluvians in the Darvaraad and smashing some of them, thus allowing them to control the flow of travel to and from the Darvaraad, as well as create choke points under their control.
Maybe the reason you think that the Eluvians have no military use to the Qunari is because you were under the false impression that there only were 4 Eluvians located in all of Southern Thedas.
The reason Morrigan probably had such trouble finding an Eluvian was because a lot of them have been moved or destroyed in the millennia since the fall of Elvhenan. Tracking all of them down would have been nigh-impossible, hence why she limited her search for Elven and Tevinter ruins that were largely undisturbed, because it had the best chance of containing an intact Eluvian still in it's original location.
Just as only a male and member of the Arishok branch can order a soldier in a military fashion. Viddasala was acting outside of her role and her gender, and even the mere fact that she was the only agent running the operation proves it wasn't official, since the Qunari would appoint multiple agents to specific roles to make sure everything worked as it was supposed to.
Once again, you are making up things about the Qun that simply aren't supported by anything established in the lore.
Sten was a soldier in the military, who was sent on a fact-finding and information-gathering mission to Ferelden.
Bull was a spy in the secret police, who was sent to lead a squad to keep streets safe in Seheron.
As we can see from the above examples, sometimes Qunari soldiers will be required to perform reconnaissance under the purview of military intelligence, while Qunari spies will be required to conduct field operations that requires cracking some skulls.
Under your belief of how the Qun supposedly operates, neither would be permitted to happen because it falls too far outside their assigned roles. So evidently your understanding of the Qun seems to fly completely contrary to how we've seen it operate countless times.