Because it won't be "better" in the sense that you really won't notice a difference in function, and because changing the full group set is a LOT more work than changing one shifter. Now, if you were going to build up a bare frame, then that Ultegra set is a good deal. But in your situation? No.

The dust caps are readily available - about $5 - if you want to match appearance.

What about your 'broke' shifter? What's wrong with it?