PS4 is the worst at handeling game instals and storage.
I bought an 8tb external for the PS4 and the task of moving everything unto it has been amazingly long. Since the PS4 only allows one external hdd at a time, I'd have to switch the two HDD I had and then wait for the game list to populate. It got pretty annoying have to do that and also keeping things in order. So I bought a new one and well you just can't simply copy over things.
What you'd have to do since the PS4 can't read two externals at the same time, is copy stuff from your old external back to the internal hdd and then over to the new external. And if the game is like over 40 gigs one copy can take an hour. And I have 176 PS4 games. So moving them all over to the new new external would take forever if I did that.
And I haven't, but it's still taken forever. I'm not sure if you could do a mix of copy and install. I'm not sure if will instal while moving games, though it will download while moving. Either way, it's just been easier to reinstall everything. Comcast dosen't have the download cap right now so thats fine. But the thing is beyond listening to the disc drive or loading up a late game save, it's mostly impossible to know if a game is fully installed. Which is annoying.
Compare this to the Xbox One, where you can have I think 4 external HDDS at the same time and easily just do a "select all" and copy over things between each other. Still can take a while, but Xbox also tells you when things are fully installed.