My answer is no, using a control pad is not a handicap in any way.
Nearly all of the sims I've played over the years are on PC rather than console, so I have almost no experience of racing on control pad at all. Indeed, as a big simulation fan who wants things to be as realistic as possible, I wouldn't even consider using any other view other than cockpit, let alone using a control pad to drive my car. However, given the massive praise control pad users get when they win and the excuses they use when they hit someone, I thought I'd give it a try to see if such comments are warranted. They're not.
As I said in the M3 @ Nürburgring thread, on my very first timed lap I set a time just one second slower than my average lap time during the race. I had no trouble at all making controlled turns, hitting the apexes, and accelerating/braking smoothly. As Hiroshi said above, even techniques like trail braking were not a problem. If I can set a decent lap with no experience or practice, then it seems clear that any lack of speed or lack of car control by control pad users is due to some other reason.
In my personal opinion, it's maybe that people just aren't sure how to make a car go fast. Many think it's a case of simply stamping on the throttle/brake at what they believe to be the right times, and trying to follow a line like they've seen their favourite F1 driver do on TV. That may work in arcade racers, but it doesn't work in sims. I would encourage everyone to read some guides on driving techniques and how to race. There are plenty of them all over the internet. Most of the techniques can be used just as easily on control pad as they can on a wheel, and you'll be surprised just how much faster you can be.
One things for certain though... "I'm on a control pad" is not an excuse for poor driving.
[As an aside, I recommend that people use the thumbsticks to control the throttle/brake. It gives you far more control and is much smoother than using the trigger buttons. It also frees up your fingers to use manual gears properly, which can significantly improve lap times on its own.]