Azure's main selling point is its integration with the rest of Microsoft's products. If you're a large non-tech company/enterprise with an IT department (all of them) that's locked into MS business/enterprise products (most of them), any sort of software development is gonna be on Azure. No other cloud can compete. I think Azure experts are already highly valuable, and will only become more valuable as more non-tech enterprises adopt some kind of internal software development team.
To the recruiter's second point: It sounds to me that, like most recruiters, they only interact with people inside the zoomer web dev bubble. Sure, MS and Azure aren't sexy, knowing them won't get you free drinks at Silicon Valley parties, but the software development world is much bigger than that.
Azure has also been way more upstream towards the PaaS market, with the focus you mention on the entire MS ecosystem. We're seeing more traditionally IaaS vendors (AWS and Google) start to move in that direction so it will be interesting to see if MS can become generic faster than they can capture specific use-cases.
To the recruiter's second point: It sounds to me that, like most recruiters, they only interact with people inside the zoomer web dev bubble. Sure, MS and Azure aren't sexy, knowing them won't get you free drinks at Silicon Valley parties, but the software development world is much bigger than that.