My wife wants me to build a new base for our breakfast table based on this one. The table isn't light, and for some reason, this design just doesn't look like it would hold a lot of weight. Any suggestions?

That's designed to look good, not to hold lots of weight. The flatter those circles are, the more fragile it will be. If she insists on that design, keep the rings as vertical as possible. If it's a rectangular table top, I'd build two bases and put the table on top, spanning the two supports and cutting the weight supported by each in half.

Can you get away with something more like this: