No it wouldn't have, it was diplomacy and luck that got us through the last mission. They knew exactly our intent and who we worked for the whole time. Ryza having a title would have changed nothing, actually they may given her some more respect, due to having earned a knighthood.
To swing back around, I disagree with this. Having a title might have made them respect her more, but it also would make very clear her exact allegiance at the end of the day.
While Ryza's aligned, her promise was reiterated in the face of them questioning what she'd do if the duke decided to be an asshole. The implication being that she as an independent but aligned operator could and would take steps to ensure their charge for free. Not necessarily with violence, but we basically promised another prison break if needed there.
A knight is expected to get with the program when ordered, and that their oaths to their lords define their primary loyalty. A knight who promises to break faith with their liege is a person asking you to trust them while actively demonstrating a willingness to be untrustworthy.
I get where you are coming from on this at least. That said I would still be very disappointed if we did not put this opportunity to use. Ryza wants to live in and repair her old home this puts that to bed quite nicely. The home situation also helps us with some of the potential manakete problems away as well, such as where they would go if they need a place to live.
All that said i'm not all interested in leaving the duchy personally. I like Ryza's relationship with Artemis, I find their interactions very enjoyable. Leaving pushes all that back. Staying means we are right here at ground zero with the current plot lines. Essentially everything is happening here so why go elsewhere.
Lets say all of the above doesn't matter because we stay anyway even though we turned down the duke, at that point we would defacto be the dukes dragon anyway. And god forbid if something happens to Artemis poor Ryza would be beside herself I don't think she could take it, so like it or not we are have already picked a side. We might as well get all the benefits.
Also if I'm reading the story conventions right. The plot is about to really kick off, and shit will meet fan. Sending us on a grand adventure gathering allies to fight the big bad.
During said adventure being Artemis' knight sounds both adorable and awesome.
Even if you want to deal with this plot, freedom in how we interact with it is still valuable. We don't need knighthood to do any of this, and taking it locks out many potential routes that we can't even see yet.
Being aligned doesn't obligate us to do what the duke and his successors say regardless of our ultimate opinion on the matter if it comes down to it. The additional benefits past the home we already have and the individual arrangements we already enter into isn't that significant relative to the price.
And again, I doubt it'd be that hard to get another offer if we want one later. Rare and powerful dragon sorcerers don't grow on trees, it'd be foolish to deny us if we change our mind later. This isn't a huge once in a lifetime offer of critical support, it's a permanent change to our situation for marginal gains compared to what we can already access.
Overall I'd rather continue Ryza's development and discovery of herself before making life changing decisions, and stay Artemis' pseudo-peer friend rather than her explicit subordinate.
Because that's an impact here too; notice the social hierarchy games we see getting played and the impact they have on the relationships that develop.
Being outside of that has compilations, but it also has benefits in how we relate to the people within it.
Edit:
You also didn't really address the the meat of my concerns.
To turn back to the wedding metaphor "But he's rich and the dress is cute" isn't a good reason to get hitched when you barely know someone. Which is basically what your argument boils down to.
Setting aside the metaphor again, this line of thinking is even less reasonable when we still have plenty of room for cooperation on other fronts.
Doing so before swearing anything binding would be good, since we'd get the direct benefits and gain an understanding of if he'll help us with what we want, how he's be willing to go about doing it, and what he might do to seve his own interests on the way.