I book all of my business travel through a travel agent and my leisure travel I deal with myself. That's because I have little margin for schedule changes in my business travel but I am willing to tolerate the risk of delays and problems in my leisure travel. It's true that, when a flight is cancelled or delayed, I could get on the phone and try to rebook myself. But it's been my experience that a travel agent can get through and get action in ways that I can't. For example, once I had a connecting flight through Chicago where my incoming flight was delayed due to weather and it was very likely that I would miss my outbound leg to London, so my travel agent said she'd stay on top of the situation and rebook me if needed. So, when I arrived at O'Hare, I was already booked onto an alternative flight to Amsterdam with connections to London, getting me to London in time for my meeting.
So, if your time table is tight, having a travel agent can make the difference between getting there in time and having the trip turn pointless. (Incidentally, given the fact that travel agents now serve mainly business clients, I think most, like mine, have a 24 hour emergency number to call. I wouldn't bother with one who didn't, since the primary reason to use them is for emergency changes to the itinerary.)
When I travel for fun, though, I'm seldom under the gun as far as timing goes. If the plane gets delayed or cancelled, it's an inconvenience, but hardly a disaster. (A quick email to my first hotel explaining that I might miss the first night but still will need the rest of my reservation is a good idea.) So I am willing to take my chances on flight problems and get into the phone queues with everyone else trying to rebook.