In an interview with DICE Executive Producer Patrick Bach, the developer admits that the announcement of Battlefield 4 has angered some avid fans and that had they announced Bad Company 3 the reaction would likely have pleased a number in their community.
Yeah, if as introductions go there was a good deal of vaguery and doublespeak, it's because Bach was being very diplomatic in his interview with VG24/7.
According to Bach, when Battlefield 4 was announced - via poorly managed PR - most of the response was "positive" but they "have some of the more avid fans who get upset." Though, he puts this down to thinking "this might mean we won't support Battlefield 3". And he has a point. Remember back when Left 4 Dead 2 was announced? Some fans formed a boycott because they worried that Valve was racing ahead on the sequel and weren't releasing expansions/DLC for the first game.
He also says "We're still working on Battlefield 3 and making sure it's the best game we have on the market right now". That's very diplomatically worded.
When asked how he thinks the community would have reacted to a Bad Company 3 announcement - something to put distance between Battlefield 3 and 4 - Bach said "We probably would have gotten a better reaction [...] But of course we've got people who want something different, depending on what you do, you always have people who want something different. That is a constant struggle".
It's clearly been something the studio has battled with internally, and the question remains: why did they choose to make another Battlefield proper?