1. I'm not sure what you're trying to say here. How is "I immediatly start hearing the calling" not a problem because I know how to defeat the darkspawn exactly? I mean that's true of the wardens right now and it appears to be a problem. Is your plan to only drink the blood before the final battle? What if the darkspawn destroy one country and we have a decades long blight? Wouldn't you then need troops who are immune to the taint to even make that possible? Except they'd all go insane in some small amount of time. This seems like a problem to me.
2. It's a matter of degree's, While yes it's going to require help from those in the area and untill then it's going to be ultimatly unchecked in a "we can't win untill they sign up kind of way" but with no one looking for the signs of the next blight that seems like a slightly worse position to me. And by slightly worse I really mean potentially catistrophic for at least one nation and potentially more.
3. So you think the countries are going to share the blood? You think if you tell them all this information that doing so itself won't cause a war for that blood when for each country this is a matter of national defense?
1. The Calling does not drive wardens insane in a very short amount of time.
The time span of DA:I is many months, probably several years, at least if you want to allow time for evacuating Haven, the long march to Skyhaven, establishing a new home, all the diplomatic messages being sent back and forth, merchants to establish permanent trading routes, alliances to be made with noble families, and, of course, all the sightseeing done by the Inquisitor), and the Grey Wardens at Adamant aren't showing any surprise that they've been hearing the Call so clearly for months without going insane, nor is Loghain/Alistair/Stroud, who after all has been a Warden for more than a decade and must have learned a bit during that time. They've got other reasons for thinking it might not be the Call at all, but they don't say, "it isn't because I haven't gone insane yet". All Grey Wardens know that it is an eventual death sentence one way or the other if it is the real Call, but driven insane in a short amount of time? If they thought that was the case, surely somebody would have commented on it when it didn't happen.
You don't need troops that are immune to the taint to fight the blight. The vast majority of troops fighting in all the blights have not been immune. You also don't need blight immune troops to wound an archdemon and get its blood, since, if you did, there would have been no first joining, due to there being nobody blight immune around to get the blood.
2. It is a tradeoff.
If nobody chooses to maintains peacetime abilities to detect the taint, they may be slower to respond, resulting in a blight that is worse. On the other hand, with everybody knowing how to build capabilities in the event of a blight, they'll be able to respond when they do act, ideally by using archdemon blood they already have, or, failing that, by planning how to get some from allies or directly from the source.
So you might end up with a blight that was much easier or much harder to deal with than the first four, depending on how exactly that fell out.
Regardless of whether any country would maintain peace time capabilities or not, they'd have a huge advantage over people during the first blight, because they know what to do rather than having to fight delaying actions against an ever spreading blight until somebody came up with a miraculous solution.
Finally, as was clearly shown with the 5th blight, that somebody is capable of giving early warning doesn't matter at all if they aren't being listened to, and that is the case regardless of whether it is Wardens or governments that have the capabilities.
3. It is entirely possible, even likely that countries would go to war over blood if it was in extremely limited supply or supplies ran out in some countries and not in others, depending on how long since the last blight it had been when this scenario occurred. If nothing else, it could make a convenient excuse for anybody bent on territorial expansion.
That is still, to my mind, much less of a risk to run than having a single point-failure in the secretive Grey Warden organization. Compared to the destruction of a blight, or the horror scenario of a blight that is never stopped, ordinary wars pale by comparison.
It should also be noted that we have no idea how much archdemon blood is needed or how much remains in Grey Warden hands at this point. They might be on the brink of running out, they might have enough to share for any conceivable future, or anything in between.
What we do know is that whatever they got from the previous blights was enough to last the Grey Wardens across the continent for at least the four centuries between the 4th and the 5th blight. Sure, the archdemons are big, but it still that suggests that very, very, little is needed per joining ceremony. Well, either that is homeopathy works in Thedas. 
(I wonder exactly what happened with the blood of the 5th archdemon, now that I come to think of it. Did the Warden and Alistair, after a manly victory shout, no doubt, cry out for buckets to collect blood in?)
We also know that to kill an archdemon in the first place you need only to have the knowledge currently in possession of the Grey Wardens and to be able to wound him, so if the world ends up in a situation where nobody has the blood, it will be known how to recreate the capability to kill archdemons by wounding it and collecting its blood or doing a non-permanent kill and collecting blood and getting it out of there come hell or high water, then using it on people and killing it for good months or years later.
If, however, the information on how to kill archdemons was lost with the Grey Wardens, it is entirely possible that the knowledge would never be recreated in time unless somebody desperate happened to have the same idea as the first Grey Wardens did, "hey, I'd wonder what happens if I mix darkspawn blood with lyrium and archdemon blood and drink it," and tried if it worked.