Yes, it is the role of CBP to make sure you're coming under the right visa. It's no different than making sure you're not coming to stay while on a temporary visa.
No whether or not this is important or just silly politics is a different question. Same to the question of is this ethical/right/whatever. But there's zero doubt that this will enable CBP to be more effective. So arguments against it should focus on "yeah, this makes their job easier, but at what cost?" instead of pretending there's zero upside.
There's still random bad luck - as a Canadian I've been theatened deportation for not cancelling my green card despite not using it or living in the US. I've been detained because a CBP agent decided a trade show was work and I wasn't eligible for a B1. (His first words were "you're going miss your flight, I'll make sure of it".) But most of the time they wave me right by.
No, but it is left entirely up to the agent interviewing you for entry to decide whether he thinks you're likely to leave later when your visa is expiring.
And is it the role of CPB to prevent under the table employment? Surely there are better ways to address that issue if it's a concern. Building walls and inspecting social media seems like a political smoke screen for avoiding more effective enforcement measures - requiring farmers/factories/roofing companies to prove their labour is legally allowed to work in the US etc. Or we could acknowledge the reality of immigration (we need workers if we can't afford to pay Americans an adequate wage to pick fields etc).