But as I made the analogy to Q, since he's a character that's part of that show's mythology and story, does that mean that it would've been OK for him to come to the rescue in any major crisis since his powers and character had been established before?
It works fine in whatever Q episode it is where the Borg are originally introduced. Q puts the Enterprise into the situation, so it makes sense that he resolve it, especially since it is a meaningful turn of events for one of the characters involved (Picard). Naturally, it wouldn't make sense for Q to keep showing up to help the Enterprise with the Borg, but within the context of that story, it makes dramatic sense.
SoA is similar, though it is taking place on a larger scale. The Prophets' intervention is an important part of Sisko's character arc, and because they are responsible for the Dominion's access to the alpha quadrant, it makes sense for them to restrict that access.
Obviously, the Prophets' influence should have its limits, but their influence is clearly defined and limited. The Federation still has to fight the war, they just don't have to fight it with unlimited reinforcements flooding into the alpha quadrant through the wormhole. If the Prophets magically brought the Romulans into the war or resolved the battle for the Chintoka system, or something of that nature, then you would have a stronger argument in my view.
The warp drive analogy is a poor one, because the Prophets are in fact currently responsible for reinforcements being able to enter the alpha quadrant (they are also technically responsible for the initial contact, but that is not the primary focus here, they are not being asked to undo what has previously been done, but to face up to what is currently transpiring). This is not something that happened in the distant past: it is happening now, and the Prophets have a choice to take responsibility, or not, just like any other character.