I can see that you're trying to explain your questions, however (a) for performance issues much more detail of exact tests would be needed (as well as a clear explanation of which figures were your test results, and which were figures from other people's tests) if this subject was actually on-topic here, but (b) this subject isn't on-topic. Questions about compatibility between different SATA HBAs and different PCs are also not on-topic (that's not a disk repair or data recovery issue).
The best picture I can get from your story, is that you were expecting a SATA3 drive to be faster than a SATA2 drive, and are surprised when that isn't true. In short, using a SATA3 drive & controller will only produce an increase in throughput
if the previous SATA2 component (replaced with one supporting SATA3) was actually the bottleneck. As far as I can see in your description, that wasn't the case, hence you saw no improvement. Therefore your bottleneck(s) is/are elsewhere. The challenge when trying any type of performance improvement, is to find the bottleneck. It isn't necessarily the interface.
I expect you'll get more help with this type of issue (and with your SATA HBA compatibility problems) in a PC-oriented forum

Good luck!