The folks who don't know that a mountain bike can BEAT a road bike for a two or three mile ride in the inner-city simply don't know much about riding a bike. The speed of a bike (until about 25mph or 30 mph) is controlled by your cadence, how fast you are spinning the pedals.
At a steady 25 mph (the speeds pro cyclists ride on a closed course) air resistance becomes the most important factor, and the lower aero postion of a road bike is a big advantage. In the inner city, with stop signs and red lights every hundred yards or so, cylists accelerate from zero to about 15 mph, hit the brakes for the next red light, and so on. Cadence control speed.
Bike messengers in Houston tend to prefer moutain bikes because of the torn up condition of Houston's inner city roads. Pot holes and broken concrete everywhere. The mountain bikes can take a straight line from A to B. The messengers on road bikes avoid the worst roads, take detours, have to dodge around the pot holes and the broken concrete, and stop and fix flat tire after flat tire. When you are paid ONLY for how many packages you actually deliver, you ride the BEST bike, and that is a mountain bike.
I ride a 20 mile inner city loop, and the times on my mountain bike are about the same as on my lightest road bike. City riding is "stop and go", and with only a hundred yards or two hundred yards to the next stop sign, it would be rare to be riding at 20 mph. Air resistance is NOT a significant factor when you are riding at speeds typical of inner city riding. But, a bike that is tough, resiliant, stable, and comfortable is essential.
The big difference? The plush tires of the mountain bike absorb road shock. The higher bar position makes it easy to watch the SUV on my left and the 18 wheeler on my right. And, it prevents neck and back pain. After 20 miles on a mountain bike, I'm ready to go on another ride.
A bike does not make you "fast". Your speed is controlled by how fast you can spin the pedals. But, riding a mountain bike makes spinning those pedals a lot more fun.