Genetically engineering and then riding a dragon actually sounds easy in comparison to creating FTL travel.
These are interesting problems to think about, right now they are solidly SF but dragons are and always will be fantasy because physics says they can't exist in the way they are described. But you can't 100% rule out that one day descendants of humanity will visit the stars in person, assuming we don't blow ourselves up first.
That's one of the main reasons I dislike Musk so much: he takes people motivated by positive dreams (a clean planet, off-world travel and living) and then subverts their energy for grift. But 'DOGE' has shown the world who he really is, the mask is off now and anybody still empowering that asshole is co-responsible.
If we assume advanced genetic as well as bio engineering that admits the possibility of self reproducing cybernetic organisms that closely match many or all of the key characteristics. I'm not sure it's actually a cyborg if the lineage is fully independent without involving any technology external to the species but I use the term there because that's the easiest way to communicate the idea.
We've already got examples of dinosaurs that come fairly close to qualifying so I'm sure something vaguely in the ballpark is within the realm of possibility. Exactly how close to the mark though I have no idea.
For example, what would you make of something visually resembling a massive western dragon but with a largely hollow body? I don't think that would disqualify it in most people's eyes. Pushing the concept a bit farther could yield something vaguely like a living cross between a hot air balloon and a deep-sea anglerfish with a weaponized tail.
Vaguely related to that hypothetical, we've got examples in the fossil record of 2 foot wide dragonflies estimated to have weighed in at under 150 grams. There's a pretty large solution space to explore here.
Archaeopteryx would have you for lunch long before you realized that it doesn't spit fire.
I'm not even clear what the criteria are. We've got examples in fiction that run the gamut. Even limited to fairly traditional post-Tolkien western high fantasy examples of creatures with a long neck, two wings, a distinct abdomen, and a tail we've still got concrete examples ranging in size from a small lap dog up to larger than a McMansion. Plenty of reasonably canonical examples don't spit fire (although being able to certainly furthers the general vibe). And weight? As I said earlier, that's (imo) solidly in the domain of secondary characteristics.
Semantic arguments aren't any fun (and are largely pointless). Is Pluto a planet or not? Unless it's a technical discussion where the distinction matters for some reason, who cares? What matters is if Pluto fits the pattern that you (ie the people conversing) care about.
Hence my posing the example of a particularly buoyant variety of dragon. I think most people would consider it to fit the pattern well enough but I'm sure there would be at least some disagreement.
Also as long as we're talking about wild far future sci-fi possibilities such as visiting other stars in person then you can't rule out dragons in the form of a wildly advanced, needlessly decadent military technology show piece. Something like a nuclear powered bio engineered cyborg for use as some sort of fighter jet. (Apologies if I just inadvertently ripped off some anime I've never heard of.)