I actually use an eSATAp SSD hard drive frequently. I have an OS installed on it and used it to work and bring it to home when I need to do home office, avoiding having to bring the whole laptop.
I went with eSATAp because:
1) My work and my home computers both have eSATAp ports
2) My work computer does not have USB3.0
3) No need for cases/enclosures to hold the circuitry necessary for USB hard drives (it is far more portable).
4) Theoretical higher performance (and lower latency) compared to USB2.0
I have not measured performance of eSATAp vs USB2.0 or USB3.0, but the SSD is blazing fast running on eSATAp. If you have more info on performance of USB3.0 vs eSATAp I am interested (latency in special is more important when running OSs on external hdd), if I ever change computers on my work I might go for a USB3.0 case.
But you are right, eSATAp days are counted, the upsides are not worth the extra port if USB3.0 can provide almost (or better) as good performance. eSATAp ports work as a USB2.0 port as well, but nobody (except me I guess) uses the SATA part, it is hard to justify their inclusion. You have no idea how hard it was to find a eSATAp cable...