I can tell you this much: It was probably done like this just to keep the plot going.

Perhaps military personal have some sort of technology that prevents them from being recorded. It might seem odd, but such technology would be extremely important; no one would want those tapes falling into enemy hands.

Either that, or, because as you mentioned, there are lots of security cameras nowadays, in the the not so far future, a state that watches every move of the citizens comes to exist. Eventually, they rebel and subsequently ban video surveillance cameras. Again, it seems odd, but this kind of backlash happens. This is why unless you are a Jainist, you can't have any swastikas in Germany, even though the Nazis were exclusively the only ones in the millennia of its existence that used it to symbolize anything other than peace or good luck. Of course who's going to tell the Germans not to ban swastikas? After the Nazis, who would blame them? Similarly, after dealing with a surveillance state, who would blame the future people for banning surveillance cameras?

Sure, it's probably just lazy programmers. But where's the fun in that explanation?

They HAD Surveilance cameras. Remember the Eden Prime video feed?

Guess Bioware's writers just got lazy.

"Ah yes, 'reapers.' We have dismissed that claim."

"But I have video evidence right here."

"It's shopped. I can tell from some of the pixels and from having seen quite a few shops in my time."

"Damn. And this was my favorite shop on the Citadel too."

So you are saying more stuff in Mass Effect should be bugged? (Triple pun)