How to transfer PS4 game files from an old internal drive to a new one (bigger one)
Solution 1:
I've done this plenty of times using the PS4's backup and restore utility with absolutely no problems. It does warn you that some games may not work, but I haven't run across this yet. IIRC trophies are the only thing that aren't transferred, but that's why they tell you to sync your trophies online first. It is a slow process though.
As pew noted as a comment, a recent PS4 update allows you to attach a drive via USB as extended storage. You can run games off the extended storage without issue.