Well, the non-Warden just handed over the cup. There was still the Warden(-Commander) present for each of them.
True, I can headcanon that WE are the ones actually doing the ceremony, that we learned it off screen between the end of DAO and the beginning of DAA. I guess Varel didn't peek when I mixed the drink. Still, I feel like I'm stretching to reach this.
Like many have said, it has more to do with plot armor than anything else.
Microbiology science is nonexistent in Thedas so we have no clue as to which vectors carry the Blight into a new host and which kinds of resistances prevent that from happening.
DAI lampshaded that some people are incredibly resistant to the taint with bad future Leliana surviving even the injection of darkspawn flesh into her body without contracting it but that's not enough to explain anything other than plot armor from taking over.
What we can explain, however, is why Bioware placed such plot armor in the first place.
Writing a game with different endings and sequels is a humongous task, the main problem is not just making sense of it all but putting that into a workable frame.
For every different option in the previous installment the next one must account for it with a diverging plot element, this means double work for the developers of the sequel and they are limited by a budget and a release date.
A prime example is Allistair whose status and romance leads to over nine different continuities in the ensuing sequels.
In Origins, having your party become tainted over time clashed with the individual character plots since Sten could not return to Par Vollen tainted, Leliana would be barred from dwelling among the elite of Orlais, Wynne was an abomination not to mention too old to survive her joining and Morrigan's gambit depended on her fertility not being hampered.
The only two characters who could become tainted and join the Wardens were Zevran and Oghren because they had already lost their ties to their previous organizations and one of them in fact joins them in the sequel.
The same goes for DA2, only Carver/Bethany contracts the taint because the story was written in a way that deals with their absense whereas other companions account for nearly 50% of the plot of the game. Even then so just Bethany's and Carver's fates generate more than six different continuities despite their lack of involvement in the main story.
Imagine the nightmare of having to write about a tainted continuity for each of your companions in Inquisition minus Solas, plot armor takes care of that.
Both Zevran and Oghren would make sense in another way...since they're set up as direct melee fighters, they would have greater chance of getting in contact with tainted blood than archers (Leli) or mages (Wynne and Morrigan). Though this doesn't explain Sten, or if Morrigan shape shifts and bites at darkspawn.
It always seemed strange that the mabari we get has the taint (except for Cousland) but by getting a flower, all is better. So no one tried to make a resistance potion/cure for the blight from this flower? Or it only works with mabari physiology?
An aside, it would've been cool if Oghren had been infected, and we put him through the Joining around the Landsmeet, or else it's slow and he shows up in DAA hoping to join as a last resort.
What, did you want the Warden-Commander standing there signing at the recruits? 
I mean...or even just have the flavor text "You mix the supah secret bloody mary and recite the double supah dupah oath and hand the chalice to X to drink." If we're feeling cheesy, we can even throw in some Ostagar flashbacks overlaid by this text.
or else...
Warden: holds up chalice and opens mouth to speak
Oghren: Eh...I thought this was some scary ceremony, not a kegger. I would've brought more bottles.
Mhairi: Will you shut up! This is an ancient tradition and a privilege only a few people ever see. Go on commander.
Warden: opens mouth again.
Anders: Last time I drank without knowing what it was, I ended up in the Pearl between a dwarf, a greased nug, and Isabella. Are you sure this is safe?
Warden:...
Oghren: Don't be a pansy! Here, ol' Oghren will show ya how it's done!
He proceeds to grab the chalice and drink, slapstick ensures. Anders and Mhairi shrug and follow suit. The commander just shakes his/her voiceless head and the scene continues.
Retour en haut
Why the hell would you actually just leave her there at the end of the DLC, what is she now, how come that she can cast "accidents" as she says, instead of spells. She is deffinately gonna come back to bite us in the ass, we have no answers to any of our questions and just her word that everything is going to be fine now 
