My understanding of Intel RST is that it increases boot speeds by writing the startup instructions to a hidden partition. This benefits HDDs a lot, but SSDs not so much.
Your R/W speeds are almost exactly on spec. The Asmedia ports if they are true SATA III ports should not be a factor. The limitation with non-native (Intel) SATA III is use as a boot drive. If you use non-native SATA III to boot there is the need to load the hardware drivers for the non-native SATA III which slows down the boot. But speed, after the driver load should be similar.
Use the magician tool to optimize the performance of your SSD. I'm currently using my 256GB 840 Pro as the main OS and using a 32GB SSD to cache my secondary drive (1TB Seagate 7200rpm). My 840 at first ranges from 300 - 530 so it's not a good benchmark. I use CrystalDiskMark and consistently get 510read/500write. With the 1TB drive with Intel smart caching off, I get about 130 read / 100 write. When on, I get 375 read / 60 write for sequential read/writes.