So, I need to clear space on one HDD, so the question is, would I re-install the games on the new HDD, then just copy the saves over to the new folder, or would I copy the entire game folder over to the new HDD and it would just work? Or would that yield registry problems?

Hard to say, when you install most games they put something somewhere in %user% with config/save information.
I'd test it out with a small game, maybe a demo. Samorost 1 is only 60MB.

Install it somewhere, open, and then close and drag it to your new drive and see if it starts. Samorost might be a bad example as it was a flash game, not sure if it has a more portable installation, maybe Bleeding Moons would be better at about 90MB.

From your message, I gather you just want to move the games from one non-system HDD to another non-system HDD? Then, if you don't want to keep anything else where drive letter matters on the HDD you're cleaning, your best route would be to just copy the files and then change the drive letter of the new HDD to that of the old (or of the partition in question, I mean), in which case as far as your system will be concerned nothing would have changed.
On the other hand, if the system isn't reinstalled, then many games (annoyingly) put their saves in the user folder, so there'd be nothing to move from that point of view.
But if you do need to change the drive letter, then the best bet would be to uninstall the games, keeping the saves (back them up first, just in case), then reinstall on the new drive and, if necessary, copy the saves over.
Also keep in mind that, if you change the drive letter, even if the game would work if simply copied, the uninstaller won't when you'll want to eventually uninstall it.

99% of the time you need to reinstall. Because logically, if you are asking this question you don't know enough about how to make the move in which you are interested in performing.

Safer way is to download the offline installers and start from scratch. Some games you can move the save files. BUuuuuuttttttttt.....not all of them. Title dependent. Make a list of your priority titles and see where those games store save files. After that, from scratch.

I would reinstall, it’s far simpler and safer and cleaner.
On an aside, I don’t know exactly when this need to have everything installed all the time came about, probably the same time as early access and clients I imagine. I always find, install only what you need at that time, uninstall when needed, and cleanup regularly afterwards. Keeps everything nice and clean, minimum space used, and less likely to break all the time. You will find the disk space needed for installer storage much smaller than installed versions, and the installers normally contain all the necessary packs the game needs to run.