This thread's methods don't seem to work on PS5. Husky's Charles method may work (see Mad Max subforum), but I personally just download the games onto my PS4 (using method #1) and then transfer them to my PS5.
One way to do this is via external storage, but this will force-start game updates as soon as you plug the drive into a console - and this behavior happens independently of any "automatic update" settings. This is especially problematic if you have really good internet - if I don't switch from wired to wifi beforehand, the game is probably updated before I can even pause it. Another way is via LAN transfer, and this didn't force-start game updates when I've used it, but it's more effort than just swapping via external.
Also, for people having issues doing this on PS4 (method #1, at least), start by making sure you're not trying to launch PSX from the download folder - I'd move it to the desktop. The PSX folder must also be whitelisted on any antivirus software. Lastly, if you don't see a green checkmark in the PSX startup window, give it a few more tries (this worked for my friend).
Tips:
Always test it by starting a download for a game you know the size of. For instance, I know Mercenary Kings v1.00 is a ~328 MB download, so if I see that, I know I'm all set. Likewise, if you encounter a connection error that pauses the download, repeat that process before resuming your primary download - sometimes it stops working and you'll need to restart PSX
The download speed is based primarily on your PC's connection, not your PS4's. I was averaging 2-4 MB/s with a wired PS4 and wireless PC, but when I switched that around, it spiked to the 25-30 MB/s range