Exactly this. In a similar situation I am aware of, a senior dev came in, found some measurable bottlenecks, convinced non-technical management that they could

Exactly this. In a similar situation I am aware of, a senior dev came in, found some measurable bottlenecks, convinced non-technical management that they could be fixed, and then fixed them. He is now the CTO. They key here is to find problems with the current environment that can be tracked. Downtime, performance, bugs in prod, etc... If you can't find any measurable issues that management can relate to, then quite frankly, there isn't really any reason for management to invest in solutions to problems that don't exist.