This means this card will not support the full 6Gb/s transfer rate of the SATA-III protocol?
No, it can, because you are probably making invalid assumptions.
The SATA interface is a synchronous channel at a fixed speed of 1.5, 3.0 or 6.0 Gb/s. You cannot slow down or vary this speed. The channel is either idle, or active and then data is transferred at a fixed rate.
The SATA interface is not directly connected to the PCIe bus. There is no stream of bits flowing from the SATA interface directly to the PCIe bus that requires the data rates to be synchronized.
Rather the transfers between these two interfaces is fully buffered. A block of command and data has to be successfully received over an interface before that block is forwarded to the other interface.
This discrete and sequential transfer of buffered blocks is common in computer and network devices (e.g. see store-and-forward switching).
Whenever there is a disparity between the speed of the receiving and sending channels, an intermediate storage buffer is the simple solution for digital data. The buffer also allows processing of the data block, such as in a HDD between the Read/Write heads and the host (e.g. SATA) interface. That intermediate (and not well known) buffer in the HDD is known as the sector buffer, and is needed to perform validation and error correction (on reads) or ECC generation (on writes) (also see When a disk read or disk write occurs, where does the data go?).
Bottom line, if the adapter claims that it has a SATA 3 interface, then that card will support the full 6Gb/s transfer rate of the SATA-III protocol.
Note that both SATA (all versions) and PCIe 2.0 use a 8b/10b encoding scheme. Every eight bits of actual data are expanded into a 10-bit code for transmission on the interface.
So the effective data rate of SATA 3 is 600MB/s compared to the 500MB/s of PCIe 2.0 1x (i.e. the speed difference is not as large as you assumed).
See What is the actual speed of SATA 3?