Azure and local number of TCP connections are very different
To answer the question itself: I wasn't able to find a way to see locally the TCP-related metrics that I get when actually running my functions in Azure. For now it feels like some important development tools and/or docs are missing. The "serverless" experience turned out be the deepest dive into Windows system programming I ever had as a .NET developer.
The solution for the problem itself was the following:I've rewritten the whole thing to be sequential and managed it to get establishing about a hundred simultaneous connections. Then I just used binary search playing with MaxDegreeOfParallelism until I found a value suitable for my plan.
You may be bumping up against the HTTP standard implementation within HttpClient which restricts the number of open connections to 2 by default. The HTTP/1.1 specification limits the number of connections from an application to two connections per server. You can override that default using the DefaultConnectionLimit property of the ServicePointManager. Microsoft has an article on it here.