I'm glad this showed up on my feed because I actually think about this a lot. So, I hope you don't mind a stranger hijacking your post for a hot second here.
I don't think anywhere in ALttP it mentions or even really hints at his age and I'm pretty sure fans outside of LU think of this particular Link and Zelda as being around 15-16. However, when you first start up ALttP, pay attention to how everyone is talking to Link. It feels like they're talking down to a child and not a young man approaching adulthood.
His uncle leaving without explaining what was going on feels like he's trying not to scare him.
The soldiers at the castle calling him a kid and trying to send him home while nagging him about staying up past his bedtime—I'm paraphrasing—telling him he could stay out later when he was grown up/an adult.
Not to mention the whole rabbit thing. In the Dark World, the golden power changes your form to reflect what's inside, hence the demon-monster bully and his wishy-washy ball friend. Small animals are used a lot to represent innocence or passiveness because how could anything so small and cute be mean or cruel or bitter or sad or rough around the edges? So, either Link is just a really passive, non-warriorlike kind soul, or he's a child.
Then we think about this from an LU viewpoint, Legend’s been on 6 adventure so far.
Now, I think it’s perfectly reasonable to assume that one adventure can last several months—I once did a playthrough of BotW just to see how long it would take to do the bare minimum without speedrunning and with prior knowledge as to where everything is; ~50 shrines, 4 divine beasts, 13 memories, master sword, and final boss took about an in-game month and a half (also, if anyone was curious the average time in between bloods moons in BotW is about 9 in-game days)—and you have to factor in the time in between adventures, the travel time to get from Hyrule to Hytopia, Holodrum and/or Labrynna + traveling between the two countries, acquiring a boat + how long he'd been sailing before getting caught in the storm, then the time spent returning home and getting a smithing apprenticeship. That's several years worth of traveling and adventuring and just life in general.
In LU, Legend is about 17-19 and if you're nicer and more generous with time than I am, that could clock him at 13-15 starting out at least.
I am not nice to Legend. I like to start him out at the brink of turning 11. It helps to highlight just how tragic his story is if you sit and think about it. He watches his uncle die, gets labeled a criminal for a crime he didn't commit, and is hunted down by brain-washed soldiers who have no qualms about bringing him back in a body bag.
So, basically, ALttP is a lot more horrifying if Legend is a child during it and Legend angst is just *chef's kiss muah!*
In all seriousness, it probably is more to give him an actual break in between his adventures to travel and learn and grow. Not counting the old man, Legend’s the most experienced hero out of the group, which means he's been at this for a very long time but he's still just a teenager.
...I think that's it. I don't really know how to end this, I didn't even mean to write this much but, well, I am me, so...