You know, I hate to put out yet another point to confuse things, but feel I must.
I'm not sure that the "power of attorney" is good enough for the permission to drive a car here, much less cross the border. I could be 100% wrong about that, but my understanding is that while I have a green card to drive the car in my own name, my friend would need a [blue?] card (can't remember what color they said it was), which does involve getting a statement signed at an escribano but there was a little more process to get the card, was what I understood. Forgive me if I'm wrong about this specific item - although I have a car in my name, I've never gone through the effort of getting permission for someone else to drive it, although I need to do this soon anyway.
The thing about crossing the border is, they are real weird about it in South America in general. IF a "power of attorney" from an escribano is good enough to drive the car in Argentina with it not in your name, I would still get in touch with a lawyer to verify exactly what you would need to cross the border with the car.
As I understand the limit on border crossings and ownership, linked with residency status (and I have had all in my 8 years of living here, having owned a car for more than 6 years and before I actually had residency):
1. If you own a car registered in Argentina, as a tourist, you should have no problems taking your car into another country with your passport, and as long as your car has Mercosur insurance. When I first bought my car, this is the status under which I crossed the border many times, into Paraguay and back, with never a problem. Until I hit #2:
2. If you own a car as an applicant for residency (you have your precaria) and have not yet been approved for residency (i.e., you do not yet have your DNI) you cannot take the car across the border, period.
3. If you have your DNI and you have Mercorsur insurance, you can take your car across the border.
When you take the car across the border in any situation, the car should leave the foreign country in which it is located within 90 days, like a person's visa (except it's not a visa, just a time limit).
As I said, I'm not sure about crossing the border if you are given permission to drive someone else's car and they're not with you, but given #2 (under which I got caught out on once, before I realized what the law was) I'd be surprised if it was that easy for a non-owner, particularly a foreigner, to be able to cross the border so easily.
Personally, I think you'd probably be fine as a foreigner, owning the car, with the proper insurance, and unless something has changed in the last 3 years (certainly possible!) you should be able to go back and forth with just your passport.
But please confirm this with a lawyer.