How can I reset file permissions back to defaults?

Read in Resolve issues caused by changing the permissions of items in .... Follow the instructions for macOS High Sierra or earlier, all the way at the bottom.

HomeBrew can't install packages because it's somehow been denied access to the folder where it stores them.

Where does it want to store them, specifically? Some file locations have been rendered off-limits in recent macOS versions, in Apple's ongoing effort to protect Mac users from themselves. Generally speaking you can do whatever you want as long as you stay within your Home folder... that's your playground.

In case it helps, /usr is off-limits, but /usr/local is ok: About System Integrity Protection on your Mac - Apple Support

I had forgotten about it, but I installed Homebrew on this Mac a while ago. It lives in /usr/local/Homebrew/

I think you need to determine the location those Homebrew packages are attempting to access.

On my system the Homebrew stuff is located in /usr/local/Homebrew and that folder has permissions 775 and are owned by myuser/admin (username/groupname), and the folders underneath it have permissions 755 and are owned by myuser/admin, but all the files in the Home-brew folder have permissions 644. So perhaps you might try:

cd /usr/local/Homebrew
find . -type d -exec chmod 755 {} \;

No guarantees this will work -- as mucking with permissions like this is not a good idea, since global assumptions are very hard to make without extensive research.

Perhaps it might be easier if you could show what the existing permissions in your Homebrew setup are:

cd /usr/local/Homebrew
ls -la

as well as tell us what the errors you're getting exactly are. Also what exactly did your "chmod" mods do?

Good luck...