I paddled for 2 seasons on the big pond (well 1.5, spent half a season getting used to kayak fishing before I ventured onto Lake Michigan) paddling a Wilderness Systems Tarpon 140. You can catch fish and do quite well paddling a kayak. However, I won a hobie at the end of last season and there is nothing that quite compares to having your hands free to fish Lake Michigan (primarily what I do) while you're peddling.
For example, if you're paddling and trolling, and hook up with a fish, especially a nice fish, you're going to need to clear any other lines you have out on the water while you fight said fish. Or at least reel them up to the point that they won't snag on bottom (lost a few spoons my first season doing this).
Also, if you want to change out a spoon, flasher fly, plug, etc on one of your rods, or take a leak, or whatever, if you're paddling, you'll probably have to reel your lines in so again, they don't snag bottom or tangle.
Now if you're peddling, you can keep a slow speed (or faster speed) while you fight a fish. You can focus on fighting the fish and not worry too much about clearing lines, unless you want to. The big difference to me is that I can keep peddling and change a line, so my lures are still in the water and give me a better chance to catch a fish.
I just caught a 10.5 lb steelhead on Friday morning that hit while I had a coho in my net, but I was still peddling while I was dealing with the coho. That's a fish I wouldn't have caught if I was still paddling.
Keep in mind, most of my fishing is trolling. Some say that trolling with your legs is a lot easier than trolling by paddling. Maybe to some extent, but I easily did 12 hour paddling sessions maintaining 2 mph in my tarpon. I'm not sure I necessarily buy that, but whatever.
If you're main focus is on trolling the big pond, I'd spend the extra money on a peddling kayak - it'll catch you more fish than if you were peddling for some of the reasons I outlined above. However, they're expensive. If you can't afford it, I'd look at some of the faster, sleeker, stable kayaks to paddle and get yourself the lightest paddle you can afford. The tarpon, kracken, thresher all come to mind.