I agree with the split. DBA = admins the database, Management = analyzes data and makes+implements decisions based on it, Analyst = support for management when specialized knowledge/skills are needed to turn data into information.

Conflating the functions is a mistake. We don't have managers (generally) who are expected to maintain the air conditioner for their office, so why expect them to admin a database or server? And we don't expect the HVAC engineer to analyze the company, the markets, the government, et al, and make decisions about the business.

I've been in both positions (DBA and management; I haven't done HVAC work in over a decade), and they require complex and different skillsets and knowledge. Managers are the ones who have to analyze data and turn it into information that can be used for planning (or reacting, depending on the urgency). Sometimes (often) they need help on that from people who are trained in areas like statistics, but that's the role of a researcher, not a DBA (DB Admin, not DB Analyst).

Those are people who deal with determining correlation vs causation, et al. It's mainly stats, scientific methodology, etc. Again, they don't need to know how to design a good database (relational or cubes or whatever), they need to know how to spot relations in complex data. "Customers who buy toothpaste also often buy dental floss" has a probable causal relationship to interest in personal hygiene, vs "Customers who buy diapers also often buy beer", which turned out to be a correlation with no real causal relationship (real-life data mining scenario). That's not something based on classical DBA skills, like normalization, SARGability, b-tree indexing, et al, or DR, backups, log shipping, replication, et al, depending on which version of DBA you mean.

Specialists cost more, but have the opportunity to be better at what they do. I view that as a positive thing.