There are two possibilities as to why the DS9 crew do not appear in the events of TOS episode S02E15 "The Trouble with Tribbles":
If we view the original episode as a separate timeline from the altered one we witness in DS9 episode S05E06 "Trials and Tribble-ations", then it becomes obvious that the TOS episode represents the original, unaltered timeline.
This would mean that all events AFTER "Trials and Tribble-ations" take place in an alternate timeline - one in which the DS9 crewmembers were present on that day in 2268.
If, however, we assume a single timeline, then we must account for how the DS9 crewmembers could be present yet not appear in "The Trouble with Tribbles". The most apparent explanation is that the DS9 crewmembers WERE there as depicted, but never appeared on-screen due to viewing angles and/or camera timing.
Given that DS9 crewmembers were digitally inserted into existing scenes from the TOS episode, we are most likely looking at the "alternate timeline" explanation. This is even hinted at in the dialogue between Sisko & the Temporal Investigators.
LUCSLY: Regulation one fifty seven, section three, paragraph eighteen.
Starfleet officers shall take all necessary precautions to minimise
any participation in historical events.
SISKO: All right. It was a mistake. But there were no lasting repercussions.
DULMUR: How do you know that? For all we know, we could be living in an alternate
timeline right now.
SISKO: If my people had caused any changes in the timeline, we would
have been the first to notice when we got back.
Although Sisko may be correct that his people would have noticed any major changes, it's highly unlikely that a minor event - such as Sisko & Dax now appearing in the background of a security recording - would be noticeable.
If you would like to see a comparison of scenes from each episode, J. Trent Adams has a page putting the TOS & DS9 episodes side-by-side. The page is old, but AFAIK it's the only one of its kind that puts each edited scene next to the original. Here is a link: