If possible, I'd like to use my existing USB HDD with all the games already installed (for PS1 and PS2).
Since you chose the second option, the process might be difficult because each model is different.
But I'll try to guide you.

Even though routers have USB ports, there's a small chance they won't support the function we're trying to achieve.
And if they do, you must take extreme precautions due to vulnerabilities, such as never sharing confidential information, securing access with a password, and only connecting the drive while playing games.
It's also worth noting that not all routers are compatible with high-capacity devices (I recommend starting with a 16GB or larger USB drive formatted as FAT32, which is generally recognized by most routers).
Will I need to buy a router, or can I use the one I already have combined with my modem?
That one will work for you.
Will the USB to Ethernet speeds change a bit?
Playing over a network is much better than using the console's USB port; you'll notice an improvement in the cutscenes.
Now, let's begin by following these steps:
1. Prepare a 16GB or larger USB drive formatted as FAT32 with the typical folders created by OPL (DVD, CD, ART, etc.) in the root directory. Place a game inside the DVD or CD folder, as appropriate.
If the game is 4GB or larger, use USBUtil, as the FAT32 format doesn't support large files, and create the game in the root directory of the USB drive.
https://www.psx-place.com/resources/usbutil-by-iseko-translated-by-jbliz7665.679/
Once you have that ready, we move on to the next step.
