Not only does the t-molding protect the edges better - it looks better. You could round out the edges of your cabinet and paint it the same color as the cabinet, but you'll be missing out on one of the major things that make it feel like an arcade instead of a peice of furniture. Everyone I've ever seen decide to skimp on the t-molding has regretted it. If your going to round the corners with a router anyways, it's just as easy to route a t-mold slot and buy some t-molding. Not only that - you can get a little creative and order t-molding a different color than the paint and have it stand out.
Take a look at some of the other cabinets around here and you'll see what I mean. Simply painting a rounded corner isn't going to be easy. Also, what kind of wood is it? Plywood rounded corners won't look good at all. If it's MDF, it will look ok, but won't be strong enough to absorb many hits.
I believe this is a time where the woodworking guy is incorrect. He may not want to buy a new bit, or he may just want to go with what appears to be an easier alternative.
Either way, let us know which way you went and what your thoughts are.