So what is your question exactly?
1) Yes, you can download PS2 games onto your PS4 running PSXITARCH Linux via the Internet or by loading them onto a USB thumb drive/portable HDD that you connect to your PS4.
2) You wonder. Good. Why don't you try yourself? PSXITARCH is just a Linux distribution. So you can install the PS2 emulator and try the games out that don't work on normal PS4 OS.
(Unfortunately the video is from a Russian guy but I guess you know how to use PCSX2?)
So basically you boot your PS4 into Linux, connect to the Internet and install the PS2 emulator.
1. Open the terminal and type the following command to install the PS2 emulator:
2. Wait for the download and installation process to complete.
3. Open the menu and go to games. There you should find the PS2 emulator.
4. Start the PS2 emulator, configure it and load up the games (from the USB thumb drive or the one you downloaded before).