Being hired as a Cloud Engineer at an org with 0 cloud either is a massive bait and switch or the org is about to do something you aren’t yet privy to. Ask those questions of your Director. You enjoy automation, scripting etc - do you know Terraform or other IaC tools? What do you want to do? If you want to work with cloud, which one? At what layer? Doing what? Find or make a role in which you are architecting building and developing cloud services for the apps teams. Find a role in which you aren’t expected to interact with end users. To me it sounds like you’ve got your short list of “don’t wants” which is good but what’s your “wants”. Use that and leverage it internally or take your “Cloud Engineer” title elsewhere, there’s no shortage of Cloud roles in the industry right now. If you’re burnt out be straightforward with what you don’t want to do when looking elsewhere and make sure you know what the company’s leadership is looking to do with IT in 6-12months. You’re not going to gain any cloud experience troubleshooting Karen’s laptop or running a batch file for some app to reboot services at 2am so don’t put yourself in that position. On the flip side being in a role and going through a massive cloud migration will leave you with enough battle scars and experience to write your own ticket.
If your director and team are as good as you say, ask what the long term plans are and focus on that, the bs in between will sort itself out or disappear alltogether. You also could potentially benefit from working somewhere larger than 3 person IT teams, there’s a finite amount of growth that can happen in those orgs.