(uh oh this became a long comment - TL;DR is that if you're finding this moment a bit cheap or stereotypical as 'man comforts broken girl', that might be on purpose)
I also agree with
@Ace435 on the nature of this situation - literally everything Iris was building up, and even things she thought should've been stable at this point in time (in particular her family name and her dad's position) all suddenly came undone, due to the recent sequence of events that she overlooked because they were unlikely enough (under normal circumstances). At times like these, I feel that the effort to maintain your composure and keep your head held up high is just too high to be able to focus on anything else, which I believe is the 'dangerous' part of this mindset that Dean is noting.
Iris in particular has it rough, because both her knowledge of herself as a reincarnation and the unavoidable rejection from the Second Prince serve constantly as reminders that she is an 'outsider'. She sees herself as alone in both her knowledge of a second world, as well as due to her supposed ostracisation in the game. I think that these events have led Iris, up until now, unable to see anyone else as a pillar of support they can rely upon, even just for a short while - and her retainers do not count, since she sees them as relying on her. See page 9 - she cannot ever cry or stop working fully, because she cannot imagine someone else taking on her full responsibilities out of no additional obligation - once again, even if just for a short while.
Dean doesn't call her weak through and through; what he is noting is that in this dire situation, it is her strength to keep pushing forward without looking after her own weaknesses which will wind up being her downfall. As someone in the royal family, he knows and has had the experience of keeping up appearances throughout his life; similarly, he admires and acknowledges Iris' strength amidst her hesitations.
And yet there's more to it than that - we know Dean isn't just there for the sake of 'protecting her'. He's there for other reasons too, and the manga makes no attempt to hide the unsettling smiles of the prince (see page 17). At least, to me, it still remains rather ambiguous on how Dean views Iris - fortunately in this case, their goals happen to align.
Yes, we end up with a scene very reminiscent of the cheesy 'prince shows up to *save* the day' cliche, but I think the dynamic between the characters (and the author's intent) is quite a bit different. Iris is for once at her wits' end, and Dean is the only one in a position to even consider asking her to 'loosen up'. We're once again reminded (and given further insight) of how alone she feels in this world - and in this one outburst, she allows herself to temporarily see the prince as someone to rely upon. (And I say 'temporarily', because literally right after she's considering that he could betray her - see page 19.) Immediately after, she shows agency by taking action of her own and calling for an opening ceremony; I don't think she's showcased as some 'damsel in distress' that must rely on the prince whenever disaster strikes.
If anything, perhaps the sour taste of this cliche scene from the slight detraction of Iris' character is even intentional. The harsh reality that some things are too tough for Iris to handle alone, and subtle hints that Dean still has ulterior motives and is getting his way, end up inverting the stereotypical associated emotions.
