The mechanic is correct. It's a little confusing going by model numbers with Shimano groupsets. The 5700 group has a system with a 1.7 cable pull ratio and shares that with most Shimano road groups up until the 11 speed generations which changed to 1.4. You need the shifters, derailleur and cassette designed for the same pull ratio. The 4700 shifter was designed to work with the newer 1.4 ratio to match the same generation newer groupsets. So, yes, the 4600 shifters will work with your current components and probably some other Shimano 10sp and Microshift R10 Xpress flat bar shifters will work.