I have a Linux host, on which I do not do AutoIT (or I use Atom to write AutoIT scripts, but I do not deploy from Linux as of now).
I have a Windows VM, and that's from that VM that as of now I deploy AutoIT scripts.
The goal is to get rid of that Windows VM and deploy directly from my Debian physical machine, to Windows physical machines in the network (and maybe to Windows VMs too one day, but not right now).
About Wine, I mostly have experience with games, and with the exception of CrayonPhysics I never was able to have one up and running...
Maybe you're right and I should try to use RemoteExec from Wine (but my chief is apparently ready to ditch RemoteExec so having a professional tool directly from Linux is probably better, if I can just identify the right one).
And yes, the AutoIT scripts we use are usually quite simple ones, like installing and configuring softwares.
And we do compile them before deployment, so Puppet or Ansible would work just fine for deploying them on Windows computers ?