You mentioned serverless and architecting for it. I consider it vendor and technological lock in with no benefits. You might have a different opinion but to each their own.
You also mentioned free tier. To me it is irrelevant as the amount of resources it gives is useless for me.
You seem to want me to defend Azure, but you're the one that brought it up. I have no prescriptive views on how to host things.
The only thing I can really offer you is that in the spirit of this article, when I did some work that would run on AWS lambda it was very handy not to have to think about any of the infrastructure that was around the business logic I needed to code.
I fully expect that code will spend its entire useful lifetime running in AWS Lambda with no need for it to escape the vendor or technology behind it. It's even possible no one will think about it until it breaks and stops sending events.
If it costs the company £50/hour for me to look into something, me being able to complete a task quickly and then never look into it again is almost certainly going to save more money than writing a dedicated process and running it on a physical server that I then need to maintain.
You also mentioned free tier. To me it is irrelevant as the amount of resources it gives is useless for me.