I've never watched babylon 5, so I can't comment too much on it.
a common trope in sci-fi is that jumpgates (or something similar) were built by some ancient super-advanced civilization, while the current civilization understands them to the minimum extent necessary to actually use them. this is a good way to sidestep the issue, since any inconsistencies can plausibly be the result of missing knowledge in-universe.
as to "whether it's still sci-fi", imo it depends a bit more on the setting. if FTL has already existed for 100+ years, it makes sense for it to be a background detail. on the other hand, if you have a story where FTL was discovered recently and it doesn't profoundly impact the plot, I would consider that to be pretty bad sci-fi.
like I said, this is just how I personally categorize things. to me, sci-fi is not just lasers in space. it's a genre where science/technology drives the plot, rather than being made up after the fact to justify the plot you want to have. it can still be sci-fi if the tech is totally implausible from a modern physics perspective; it just needs to be consistent.
star wars is usually my go-to example of "not sci-fi". there are no consistent rules for how anything works in-universe. all the technology works in whatever way is necessary to justify the current scene and will likely contradict itself later for a different scene. it's still a very enjoyable franchise, just not what I would consider sci-fi. a show like the expanse is almost the opposite. there's some unrealistic stuff in there, but it's internally consistent. once you nail down what's possible given "the rules" in-universe, the plot almost writes itself.