Joe_OC, you seem a little bit inexperienced to be suggesting something like this. We use a mix of both systems, it depends on what is in question. The first problem is that scientists reclassifying species is an ongoing process. Common names on the other hand are more stable. Secondly, a lot of things are unidentified. You would have a lot of different "Garcinia sp." and "Plinia sp." to disambiguate. Thirdly, as has been mentioned here, there are plenty of hybrids that don't always have a neat new binomial, especially when they can be F2 and F3 hybrids. I have asked botanists for the binomial of boysenberry and never gotten a consistent answer. Citrus and dragonfruit are quite messy, and you can't just use the genus name. Fourthly, there are a lot of repeated species names used in many genera. Take names like hispida, cuspidata, macrophylla, chinensis, indica, armeniaca. Saying the full binomial takes more syllables than is practical.
Think of it this way, you are never going to call a grape Vitis vinifera, or a peach Prunus persica. Once rarer things become a bit more established, the common name naturally takes over. Having two systems is more of a help than a hindrance in my view, both should be used in similar frequency.
Having said that, it is a problem that there can be too many common names that are synonyms - it's just something you have to deal with. Can you believe that there are still people who call carambola the "Five corner fruit"? At least star fruit has some imagination. My recent pet hate is hearing "Beach cambuca" for Myrciaria strigipes, when it should be Beach cabeludinha. Appropriating names for things that are not so closely related is dumb in my opinion. Indigenous names are becoming more popular, though they are often a mouthful and too repetitive as well. The best from each world will win out long term.