Impact of Playing Multiple Hands?

To a non-counter, is there any value to playing multiple hands at once with flat betting? (So not playing more hands on a good count, or depth charging.)

I'm inclined to say that there isn't since each hand would be regarded independently against a single dealer's hand, exposing more money to the edge (but not more than playing the same amount on a single hand) and simply amplifying the effect of variance. However, in practice, I seem to end up on the positive side of things more often than when I was playing a single hand.

Your inclination is correct. You are always subject to the house edge therefore if you increase the amount bet you increase your theoretical loss as well.

You can experience less variance by halving your bet and playing two spots. Instead of one hand at $50 play two at $25. The decrease in variance comes because the hands are not independent as they're both played against the dealer's hand. There's still that darn house edge though.

Note that some casinos require you to bet more when playing two hands. I'm assuming you are playing shoe games. In pitch games you have to play your first hand before looking at your second one.

It's the commonest trend here in Asian tables to play multiple hands, some even up to 5 or 6 boxes, then plus splits and double down. Basically, we see this is a better way to "Hedge play" as the Dealer may not hold 1 or 10 all the time and if faced with such high 9s or 10s, we simply surrender and cut the losses on bad boxes. On the long run, Dealer pays out more often and players lasts longer or walk out faster on short wins.
And oh, Card Counting died with the implementation of CSM. The continious shuffling machines killed card counting and it's ridiculous to say anyone can still count.

Best of luck!