I'd argue you need this split - managing people is different skill set to managing a project, which is different to managing the technology, which is different to managing a product.
They do overlap, but to me there is a clear line between their responsibilities and authority. Things get blurry in smaller orgs, but in larger ones with multiple product streams, budgets and stakeholders - they're usually pretty clearly defined roles.
Like many things, it sounds good in theory which is why many try it... but in reality it often turns to shit because you end up with 4 quazi-managers all stepping over everyone else and constantly asking the team to do extra work (like generating data/reports or attend meetings) that's really their own damn jobs because surprise-surprise being technically capable of more than making spreadsheets wasn't part of the original job requirements.
They do overlap, but to me there is a clear line between their responsibilities and authority. Things get blurry in smaller orgs, but in larger ones with multiple product streams, budgets and stakeholders - they're usually pretty clearly defined roles.