I think Tim largely did not see himself as a kid. Definitely so by his first loss as Robin. He called his peers "kid" and yes that's cringe but I think it also has to do with how he saw himself.
Robin cannot be a kid and Batman's partner and offer the grieving Bruce what he needs.
Tim, who was neglected even if it wasn't as intense as in Fanon, doesn't even really know his parents and didn't get that good relationship building with even one of them until the end of Dad's life, so he did not have that reliability.
He treated himself like an adult. He was constantly going on his own missions as Robin and planning things out and informing Bruce as needed. He full on had plans to "take down Bruce if needed" before he was seventeen. He was fully independent much, much too early and even when he first lived with Bruce it was in a guest house at thirteen.
Tim in a way differentiates himself because even as Tim he is not a Kid. He is Mature and can be left alone. He takes care of himself.
I think that, yes, this is a good thing to explore but I also think this positioning himself as other does very firmly tie into why he is always discouraging other kids his age an younger from being vigilantes so often. They are Kids, not like him. They didn't have to step up and be Batman's support. A position he took because he felt it was Needed and he has multiple times expressed a desire to retire from until after the Worst Year of his Life.
Real kids need to be protected and live nice normal lives because Tim is viscerally aware how horrible it is, both from his own experience and watching the Bats. Real kids don't deserve to have to go through that. Yes, it does come across as arrogant and asshole-y and annoying as fuck and dismissive and tone deaf as hell. Especially when directed at like Steph but more understandable when directed at someone like Dodge. Especially given Tim has been part of two teen superhero groups.
But I think Tim genuinely sees it as preventing someone in the category of "needing to be protected/has earned the right to be protected/is a true child" from being injured. Steph deserves a normal life and doesn't have to do this. Not like Tim. He is trying to be kind and stop them from making a mistake (like he may sometimes think he did) and devoting themselves to something that will hurt them and get themselves or their family killed.
Tim, of course, is not in this category. He is Other. Much like he's always been. He doesn't really start to view himself as maybe in the category of child again beyond needs and branch out briefly before Damian arrives/Bruce dies following his adoption.
But then Bruce "dies" and he becomes Red Robin and he has fully become the Other. Even Tim's retirement isn't possible anymore. He's going to be a vigilante for life. And he loves the Bats and what he has done and his friends and admires other heroes and even loves this life and loves Robin, but he wouldn't wish that on anyone.
