My computer was built in 2017 and I'm going to build a new one. Can I just move the installed directory and run the games without installing them again. I use the offline Installer and don't want to have to install over 700 games. I would take my weeks or months

Maybe! It depends on the game. Some of them stupidly tied their settings into the Windows registry, while others literally do not give a toss where you put them.

Edit: Maybe if your machine is that old, you should consider Linux to save the trouble of Windows.

Someone with greater technical knowledge will no doubt answer this in more detail than I can, but based on my own past experience, you will have to reinstall SOME of your games, but not all of them if you do the copy/move process. I think it has to do with the system registry, but I'm not certain of that.

One alternative that might work would be to use a tool to clone your hard drive to the new one if you're not upgrading the operating system. That may be the route I take myself whenever I manage to replace my ancient system, since when I do it won't be with a NEW system, but another used one that's simply less old. ;)

I don't know if that would work. Last time I tried it (over 15 year ago) it was either windows 7, XP or 98 don't remember which. I got the blue screen because the drivers for the old computer weren't compatible with the new computer and it wouldn't boot. Wondering if windows 10 is different?

I honestly don't know for sure. My nearly 15 year old system is still running Win7, so while I have used Win10 in other places, I'm less familiar with it. I'm not real knowledgeable on the technical side of these sorts of things unfortunately; I mean, I can follow instructions fine but I don't always understand WHY something does or des not work. My guess would be that as long as Win10 supports your new hardware (and vice versa where necessary) the cloning would work, but I wouldn't give it a try until someone who has a more solid understanding of this stuff chimes in.

I have too much software/games that won't run on Linux

Just try. For most GOG games it does work. But not for all. Especially some of the larger AAA games have to be installed 'correctly'. I used this method to migrate games to a computer where I have no admin rights and cannot install stuff and for about 80% of my games it worked.

From my own experience: Yes for most games, with certain reservations. Like Witcher 3 is a game I haven't installed for several reformat of my harddrive. But yeah, easier to just say to try each one.

Though, while the game itself might just re-create the necessary register info, you have to at least install the external libraries that usually comes included in the games installer:
https://www.techpowerup.com/download/visual-c-redistributable-runtime-package-all-in-one

https://www.techpowerup.com/download/directx-redistributable-runtime/

These are the bare-minimum. I install those packages every time I reformat and reinstall Windows. There's a wiki out there that can tell if a game needs more. Most don't.

Also, you'd be surprised how "portable" Windows can be but you may need to re-reactivate depending on the version you have. I have often from XP to 10 installed Windows on an external HDD and have it boot up on another computer, say between job and home computer.

Now, more than often, with graphics card, net card, and going from intel to AMD computer and vice versa, is going to complain about lack of drivers, but it usually won't give you the blue screen (due to standardized fallback drivers that Windows comes with). Same thing with the PE version, you can anticipate the need for a driver and then either slipstream it before hand or just install the drivers on the live system before moving it to the new computer.

Just remember that most games released after (roughly) the middle '00s don't save most (or any) of their savegame and configuration data in the installation directory, so you'd still need to find where each game saves its data to -- your best bet for this probably being PCGamingWiki -- and then transfer that data to the corresponding location on your new system. Assuming you care about transferring your progress and settings (and that you don't already back up this stuff regularly), that would/will be a much bigger headache than merely getting the games themselves installed.

Honestly, though, I don't know why you'd possibly need or even want 700+ games installed at once. Just bring over the ones you know you want to play imminently, and install (or copy over) the rest as you feel like actually playing them. (But DO make sure to back up any of that save data you might care about from your old PC!)

Maybe if it was Linux, you wouldn't have to, as the whole program is in the shell.
But in case of Windows, it depends on the structure of the program, but rather you should install games again.
In most cases when the game is installed, it creates the entries in the register, caches or place for saves in documents.
Basically all additional files, which are needed for the game to succesfully operate.