In fact, the eSATA spec does NOT provide power to the external device through the port, and so almost all eSATA devices have their own power supplies. There is a newer version of eSATA (not "standard", I believe) that adds some power supply contacts in the port, but it's uncommon.
The two more common features of eSATA that MAY not be supported in standard internal SATA ports are hot swapping (you say you don't need), and longer cable lengths to reach an external device (done by higher signal levels). However, many internal SATA ports actually have added those eSATA features, so often the adapter plate you mention works just fine. Just don't push for an extra-long data cable to the external drive.