Yes, I'm aware of how time travel works in the trekverse but we aren't given any reason why Sisko & Co's incursion here could cause the elimination of Star Fleet. In City they gave us something - the Nazi's won, blah blah. Here, nothing.
They left Bell's body there for the police to find. No reason Sisko's picture should have been on the ID.
(Time travel hasn't changed in the 2009 movie either, they just don't want to admit it.)
We kind of get left to fill in the blanks, but it's not hard to see how the bell riots could end up being a greater civil uprising, or part of one, that could lead to all sorts of ripple effects further down the line.
And yeah, the 2009 movie....well, I guess time will tell, though it goes out of its way to try to sleight of hand it in dialogue, and then there's the fact that the universe hasn't gone bye byes because of the huge paradox yet (which may also be am explanation for the starfleet is missing in Ds9....the universe is a storm of booming paradox, with the instigators being the last to go at the eye of the storm. After all, if starfleet/sisko/red matter jellyfish never exist, the timeline can never be altered, so they do exist and alter the timeline which means that never exist....all together now O'brien fans....)