Really, your post answers your question. There is value (and risk) in either specializing or generalizing, and there is room in the world (and the job market) for both.

Personally, I am a database specialist. I do have skills in other areas (I do python coding as a hobby and I can write C# code that at least works), but my strong suit is definitely in databases, and even with databases I focus on MS SQL Server and Access with just a little experience in some others.

I personally chose to do that because databases appealed to me and by specializing I could gain a depth of knowledge that very few generalists can. In terms of careers this limits my options somewhat, but it means that within my particular domain I can out compete most others. When the company needs a generalist that can do a lot of things, I am probably not their best choice, but when they need someone with a deep understanding of SQL Server and relational theory then I am a good candidate.

Its all about trade offs really.