I've lived here for three and a half years and have never been stopped for a police check. In the last two years since I've had a car, I've driven by dozens of roadside checks - monthly - and have never been stopped.
Even if you are, I don't believe they are really going to care about your visa - the local police don't enforce immigration issues, I don't believe. A sharp one may use it to get a little "forget it" gift from you maybe.
I've booked plane flights and bus trips without my passport. I've only needed it when I boarded international flights. Trips to Cordoba I never had to show my passport once.
And I've had tremendous luck using my Texas driver's license when using my debit cards. I stopped carrying my passport three years ago.
I have had a couple of issues in CompuMundo out here in Pilar. After about a year of having shopped there on and off, one time I tried to pay for something with my debit card and whipped out my license like usual. The clerk told me that I had to have a DNI or my passport and I argued with her a bit about the fact that I have never had problems with my TDL before. She ended up letting me use it, then told me the next time I had to use my passport. The next time, I had the same problem, but this time while I was waiting in line, I noticed there was a sign taped to the counter that informed customers that DNIs, passport, cedulas and other official government-issued id cards with photos could be used. I pointed to that when she told me I had to show my passport and she accepted my TDL with no more argument.
Easy is the only place where I have ever been denied using my TDL.