The cure for the Calling already existed long before the HoF started searching.

I think this is one of those plot details that wasn't thought out that much and then conveniently forgotten about. There wasn't really any particular reason why either foetus would have been infected by the taint. After all, if the semen was infected then surely Morrigan would have been as well? However, for the dark ritual to be possible they had to say that the baby had inherited the taint from the father but if the ritual also cleansed the taint then it would no longer have attracted the soul of the old god. In fact, if you think about it carefully, if the old god soul was only attracted to the foetus because it was infected, then surely the old god is infected at a spiritual level rather than just a bodily one? If Flemeth had access to magic that could completely cleanse the baby and the old god of the taint, why was she keeping it to herself?

So actually a better explanation would be that the magic simply attracts the old god soul to the foetus, for whatever reason, that is able to overcome the normal soul jump process.  For example, Morrigan having been the child of Flemeth/Mythal may have some aspect to her physiology that makes her more like the ancient being that is the arch-demon.   So all the warden is doing is enabling Morrigan to conceive under circumstances of the dark ritual which sets up the beacon that makes the old god soul aware of it. Of course, if that were the case, then it wouldn't be vital that a warden was the parent, simply that she undertook the ritual at the required time.  So there are still paradoxes in this alternative explanation.

Really, just as with the time magic plot in DAI or the taint cleansing Fiona in the Calling, thinking about the details just raises more problems, so as the writers  say in the objection you can raise to the time magic assertion,  it is a case of "it's ...... magic, go with it."

Avernus discovered a way for a Grey Warden to live for hundreds of years beyond the average human lifespan. The way I see it, that blows any possible cure for the Calling well out of the water.

Another instance of the writers fudging the issue. Now you could argue the Wardens only recently became aware of the success of Avernus' research because for many years than had been banned from Ferelden. However, once they were aware, given how hard it is to find recruits and the wastage level of the Joining, using his research would seem to make sense, even if it only prolonged their lives by decades rather than centuries, unless there were strong ethical reasons not to. Bearing in mind that Clarel and Co seemed to think the sacrifice of their fellow wardens was justified in the fight against the darkspawn, and they have never objected to the use of blood magic in the cause, I do wonder why they didn't follow up on what the Hero of Ferelden discovered from Avernus.