It's a mistake. With it's warp core powered down Discovery should have had only impulse. Someone in production forgot or overlooked or simply didn't care what a "realistic" amount of time would be to get from Jupiter to Earth on impulse only and therefore the ship traveled at speed of plot and pacing. And travel at the speed of plot or
https://tvtropes.org/pmwiki/pmwiki.php/Main/ScifiWritersHaveNoSenseOfScale are time honored Star Trek traditions. It's a goof. Ignore.
Other than that: 1) Since this is likely a mistake how much the Federation improved it's fusion technology should be dismissible, because Discovery's speed is too high for the situation described and we would be comparing DS9 with literal nonsense numbers. 2) See number one. 3) That's one of those things that I would be very happy to see ignored and left to the likes of Dr Who, but if we have to handwave it, that technology was somehow one of the associated technologies to time travel and was lost with time travel technology. 4) The fusion reactor don't power the ship's warp directly. It just the first step. Federation ships, and the ships of most other species, need a warp core to reach faster than light travel. We know that the energy needed for sustained warp comes from controlled matter-antimatter reactions within the core by smashing deuterium and anti-deuterium together. And we know that dilithium is the stuff that can interact with and regulate the anti-matter used in warp cores and turn the resulting energy of the matter-antimatter annihilation into electro-plasma. It's essentially harnessed energy and can be transferred by plasma conduits (and eps conduits to other ship systems) and is what ultimately makes the ships go wroom wroom. The fusion reactors for impulse are also creating plasma and can propel the ship, but even in the TNG era max impulse was "only" a quarter of the speed of light, so I'm not sure if simply propping up some more normal fusion generators will do the trick here, since I'm assuming that the plasma it generates is not nearly as highly charged as a warp core output.
That's what they need dilithium for. You can get to at least warp 1 with normal nuclear fusion. Whatever Zephram Cochrane used to create warp plasma (from what BTS stuff I remember some repurposed warhead) got him there. But that was a one minute flight, and we have no information if the ship would have been capable to sustain that for any lengthy amount of time. What we do know is that Federation ships need warp cores, or a lot of plotlines from the older shows would have stopped after three minutes with, "Oh, warp core's damaged? Just hook up the impulse reactors and then onwards at warp 3".