Automation doesn't fix a broken process. It runs the broken process faster, more often, and with fewer people watching.
The pitch is always efficiency. Take the thing your team does by hand, hand it to software, and reclaim the hours. When the underlying process is sound, that's exactly what happens, and it's a good day. The problem is what happens when the process underneath was never sound to begin with.
If the workflow has unnecessary steps, automating it preserves those steps in code, where they're harder to see and harder to question. If approvals route through the wrong people, automation routes them faster through the wrong people. If the data going in is poor, you now generate poor output at machine speed. Nothing was repaired. The mess simply got a budget and a dashboard.
The unfashionable step is the one most worth doing first: look hard at the process, throw out what shouldn't be there, fix what's left, and only then automate. It's slower, it's less impressive in a steering committee, and it's the difference between automation that helps and automation that scales the original problem.
Automating before simplifying is paying to do the wrong thing more efficiently.