As with most things the answer is yes for some, also no for many more, also it depends (some things will work fine, others less so), and just for fun it will also depend upon your technical capabilities and/or willingness to learn. I am going to assume you have limited bandwidth (be it download cap or just slow) but still some and a lot of time for this one.
If you can clone your old hard drive to the new one then that would be great, assuming Windows will boot. Some serials and DRM might be tied to hardware IDs so you might have to log in to a service to say I am good or otherwise crack the game, to say nothing of the same thing probably happening for Windows.
There are games that run from their folder, and maybe have a save in My Documents or whatever. These you can copy around. These are mostly just DRM free games though and not all of them. When you installed them if that mostly meant extracting a zip file or something then that, most others will have things elsewhere you need.
Most games however will probably be installed. This means some files in program files (or program files X86), some data in the registry and possibly some other data in the local settings/application data folder, maybe something in the windows sub folder (DRM, anti cheat and what have you). You can try to find everything (if you have an installer then that will tell you, however that would be besides the point, if you back up the drive, uninstall it, note what changed and restore the hard drive so you can copy it all back then OK) and make sure it gets copied to the new place but still expect some fallout.
For Steam there are also things like
https://www.traynier.com/software/steammover but that is more for stuffing games onto larger hard drives to leave a SSD clear, still can help.
Steam does have a feature like this but it is limited in what games will play nicely with it
https://support.steampowered.com/kb_article.php?ref=8794-yphv-2033
You can possibly also abuse Steam itself by copying files around and having it redownload the handful that might be messed up. Torrent based installers will probably also be good here.
Going back to the clone the hard drive there are ways to turn an existing install into a virtual machine. Not ideal for games but modern VMs can speak directly to the CPU and GPU so maybe something could happen there.
Short version. It has been the better part of 3 decades since I first installed an OS on a computer and started fixing them/fiddling with them. I have not stopped since. I have pulled things back from the brink many times and fixed machines. Even with all that said I would probably still just grab any saves and reinstall/redownload as doing more than about 2 games that want to make life a pain will be super annoying.