I'm late to this party... blame work and my lack of ability to think of a friggin title for my next book. But I have thoughts!!!
I can see where your concerns are coming from, but I am going to wait and see how the movie turns out before I criticize any actual plot. Half of the time, the trailers are cutting random scenes together to look like one flowing story, but it doesn't reflect the actual product at all. So, we'll see.
In a lot of versions of Batman, he isn't played as something as public as Superman. He is a legend within Gotham that people know about, but never really know for sure. So if this is the version that they're going with, then even an older Batman could be big news, as long as this is the first time he's shown himself to the public.
As for how well the press covers things... In the real world, there are plenty of issues that the press dances around and avoids all the time. It's the difference between a "mass shooting" and a "terrorist attack". Referring to a "bat vigilante" paints a very different picture than "Batman". So basically, I'm saying that when it comes to the press, I don't necessarily look for or expect common sense, logic, or reason.
The trailer was interesting, but it confused me. And I'm not sure that it's painting the right picture. This Superman doesn't look like the character from Man of Steel. He is angry and bitter and apparently has an army of Nazi soldiers bowing before him. I want to get a sense of the character from before. If he's changed, I want to know what happened. I get the intrigue of presenting him this way, but that really only works with the trailer from Batman's POV and I think it only works if there is another trailer from Superman's POV. This is a sequel to his movie, after all. I feel like they're making "Batman" trailers because they think there's more of a market for Batman than Superman. But people want to see both of these characters equally!
Batman in the trench coat. I've seen that before, but I can't remember where exactly. Damien as Batman? Or is it just ringing a Hush bell? Is this really Bruce Wayne? It's a cool image though.
The voice is Batman. It makes sense that he would use it even around people that he knows, because he is Batman. Unlike Superman, Batman is closer to his true self than Bruce would be (in Batman Beyond Bruce even thinks of himself as Batman in his own head).
The trailer overall just confuses me. I'm excited about the movie, but the trailer didn't give me a sense of what that movie is actually about. I find it kinda frustrating.
And Lex... I really hate the way he looks in the trailer. The portrayal seems completely wrong. He looks stupid. His voice is annoying. The question is, is this just his costume?
Wonder Woman. We didn't see much, but what we saw was pretty cool.
I am tempted to compare this to the Civil War trailer (which just feels like a response to BvS to me), but another part of me doesn't even want to go there. The two companies aren't even really competing at this point. They're playing completely different games, with completely different rules. I'm sure there's something worth talking about with Civil War, but if I do that here it could just muddy this conversation.