So for anyone struggling with the helper tool app insisting that Arma 3 Tools isn't installed, I looked at the source code, and the app seems to be digging around in non-existent registry entries to find the installation directory. In my case I had to add the install directory to a string entry here with Regedit to fix it:

I'm not sure if Steam simply neglected to create the registry values or what, but I suspect it's because I copied my Steam library from my old PC. Completely uninstalling and reinstalling Arma 3 Tools might fix it, but verifying the tools files clearly does d*ck all, in classic Valve fashion.