I not saying they would ignore everything that makes a warden a warden. Remember now, these warden of today are just people passing the rituals and knowledge of the wardens to the next generation till it's needs. They just do that while tainting themselves as well. All I'm saying is that they should do that with out the part about tainting themselves.The order stays, everything else stays, just not joining unless needed. I don't see how the only way info on wardens can only be passed if everyone in the order has taint in them.
I've had similar ideas to yours. Maintain a relatively large number of non-tainted recruits, with tainted officers training them and leading them on campaign against the darkspawn in the Deep Roads and Anderfels. The knowledge of how to become a Grey Warden would be known among them all, with several discreet other groups (First Enchanters of Circles, who provide the magepower necessary for the Joining, as seen in Ostagar) and cache also containing the information, to prevent a near wipe-out from destroying the possibility of Wardens.
The thing is, the Grey Wardens are lacking in manpower.
Wardens are needed to stop a Blight, but I see no need to Join every recruit that wants to fight darkspawn and be prepared. Normal humans often fight darkspawn. Not quite as well, and they're at risk for infection, but at least they get some use before they die, unlike if they fail the Joining. In the event of a Blight, everyone is Joined, and mass conscription begins with general approval. This would avoid wasting the precious Archdemon blood during non-Blight timeframes, which can stretch for centuries.
There's logistical flaws in this, of course. Why would people be more willing to be recruited into the Wardens without a Blight? They're not flocking to the Order when all they have is rumors about high casualty rates for recruits. The non-Joined wouldn't be able to slay Archdemons or sense darkspawn, but they could man a garrison.
Part of what sparked this idea for me was the Teutonic Order. The Wardens would be the actual members of the Teutonic Order, and the non-Joined would be the various men-at-arms, knights that came there for glory, and peasants living on the farmland near the castles. (Though Wardens don't generally have that level of power outside of the Anderfels.)
That's part of what I would do if I was the First Warden, and the Wardens were a centralized organization. However, it appears after the Fourth Blight, they lost all but the pretense of the chain of command, with every Warden-Commander largely deciding matters for themselves. And Warden enthusiasm has never been high enough for them to be a full army. The griffons allowed them to be a highly specialized force that coordinated the other nations into cooperation against the Blight by force of personality, legend, and offering the only hope to take down the Blight. Also, griffons. 
And I'm definitely in agreement on the neutral thing. The only time Wardens should ever be involved in national politics is if they have to gain control of a situation in a Blight. Origins is the hilariously out-of-control-situation that all other FUBAR's in Blight history are measured by. 
But as seen in Inquisition, a powerful military order that answers to no government can quickly get out of hand. It's a wonderfully awkward middle ground between not becoming aligned with a government to avoid conflict of interest, and courting them to make sure that they can actually make use of their troops.
I'm very curious as to what's going on with the First Warden and his suspicions of blood magic manipulating the Wardens, and the apparent schism between Northern Wardens and Southern Wardens.