The downside is that if the existence of dragons gets out, Ryza will almost certainly be hunted. Her stone alone is an utterly priceless instrument of magic, probably the reason humans hunted down all the other dragons originally. Recall the way Artemis goggled at Ryza having an entire stone. Even the shards are apparently stupendously valuable.
Okay, first off, have you heard of sarcasm? Second off, that's only a risk if the people run into Ryza in dragon form, which definitely won't happen with the Mysterious Stranger event and might not happen with the Pegasus Knights one. (Probably will, but maybe they won't.)
Second off, my estimation of the threat posed by wannabe dragonslayers is
considerably lower than yours. First off, in a pre-telegraph world, news of this dragon-girl is going to travel slow as molasses, and it wouldn't get very far before becoming indistinguishable from whatever the local equivalent of Bigfoot sightings is. It would take a consistent pattern of sightings before anyone would dig out their Wyrmslayers over it...and before it got to that point, Ryza, Artemis, and anyone in Artemis's family who she told would also hear about these rumors, so we'd have time to prepare.
Which is important, because that preparation could easily involve a certain quasi-royal family announcing that the dragon is under their protection. I find it plausible that dragonstones would be "mess with a dragon" valuable, but not "mess with an archduchy" valuable.
Finally...this is a story. My choices, as always, are influenced by what I think would make for a good story. Meeting some new people makes for a good story. Ryza needing to hide her dragon side makes it more interesting. And breaking the Masquerade, while unlikely to result from this one meeting, could be interesting—or at least could prevent said Masquerade from growing stale and annoying.
Dragons have four legs, wyverns have 2. I don't think pegasus knights who are likely to have seen actual wyverns will mess it up while they're up close.
I'm also very worried about them attacking what appears to be a beast kidnapping a girl. And if Ryza has to speak to get them to stop that really blows her cover.
I would think pegasus knights who have seen wyverns (and probably wyvern-riders) up close would be able to tell the difference between someone
being kidnapped by a monster and someone
riding a monster.
And while Ryza talking would both explain the situation and blow her cover...Artemis can
also talk. "This is my noble steed, look how 'tame' she is" dispels those fears just as easily.
The purpose of heading to Agrithe is to safely get Artemis to a place of safety to be able to regroup and do something.
Exposing Artemis to a host of Pegasus Knights doesn't serve that aim.
So, you think that a
complete stranger is less likely to endanger Artemis than some pegasus knights? Do you think she'd be attacked by pegasus knights from the Whitewings, a group Artemis was going to ask for help, or that there's a group of hostile hitherfore-unmentioned pegasus knights who might show up and attack?
Especially because there is no way you could pretend Ryza is a wyvern.
Unless you count "comically failing to convince them".
I mean, I'll play the math game if that's what's really needed to keep things semi-realistic, but it's not my favorite part of story writing. Maybe that's a weakness of mine, maybe it's not, but it's where I'm at right now.
I think it's less a matter of realism, and more a matter of trying to figure out the actual costs and benefits of each course of action. The costs of taking the faster route are clear—no chance to read, plus an interesting encounter—but the cost of taking our time is
not. Are we saving half a day, or half a week?
Now, I'm interested in the encounters for their own sake and consider the saved time to just be a bonus; I don't care much about the exact number. But it seemingly does to other people, particularly who are on the fence.