When I was in Dothan, I would VT the first and last hour of my show. Then do 4p-6p live. I would give temps. "Sunny, hot and 91...here's Tim McGraw". (It's not a stretch to look at the weather forecast and make an educated guess)
Outside of the morning show, the relevance of the time is almost nill. At the top of the hour you can get away with: "It's 4 o'clock!" even if it's 4:02. People aren't thinking, "4:02?! I'm going to be late for work!".
I know plenty of other jocks will bank requests and then play them back when the song comes up. "96.9 The Big Dog, that was Blake Shelton, "CAN I HEAR THE NEW ONE FROM RASCALL FLATTS?"...right up the post. If you put in the effort, then the listener will never know.
Speaking of: It's not that DJ's are trying to deceive the listener. It's a man-power issue. It's the state of the economy and of radio as a whole.
Back in the day, before computers or automation, you had to have a human being in the radio station. Now, most jocks have secondary duties, and voice-tracking is the ONLY WAY they can finish up their busy work. For example, in Dothan, I was Chief Engineer + IT guy + Afternoon host. It would have made for a very long day, had I had to spend an hour prepping and then 4 hours in the studio.