I feel the Playstation TV is underpowered for a mini-console these days, and rather get the Nvidia Shield TV which can play console emulators, Android games, stream PC games from a gaming PC, and use Android TV apps.
Those Sony Memory sticks are also kind of expensive. The Nvidia Shield TV has a 500GB hard drive version, or you can plug in a USB storage drive to install more apps, and games. The 2016 version of the Shield TV has a MicroSD memory card slot, but the 2017 version does not have MicroSD support. But, you can use a USB storage drive to add more storage on the 2017 Shield TV.
I'm not sure if there is any emulation for Vita these days or not.
I bet, if there was emulation for the Vita, the community of emulator coders/programmers is less active than other emulator communities for emulators for Windows, Linux, and Android which has a lot of gamers.
There is always always a chance that Sony could release a software update to block emulators on Vita, so using a Vita for emulation may not be the best choice. Sony release an update for the PS3 which blocked Other OS on the PS3 which made it harder for users to install Linux on their PS3's hard drive, so I can see Sony releasing updates for the Vita to block emulators from working.
I didn't even know there are PlayStation TV's out there. I might have to look into getting one. Especially since they are mini consoles which I like better than the larger ones.