Just over 7 years after the show debuted, I think I've figured out what the Blue Lion's purpose in the team is.
While at first glance, Voltron seems to follow the five man band trope (Leader, Lancer, Smart Guy, Stong Guy, Heart) pretty closely, color-coded and all.
Black is the leader: cool under pressure, and always able to see reason and the best course of action, and Shiro fits this perfectly.
Red is the Lancer: the leader's right hand man, the one to challenge them, the one who is almost as good as them, and the character foil for them. Keith matches up with this as well.
Green is the Smart Guy: the one with knowledge specialized to the setting, the one the team always looks to when they need something niche, and Pidge/Katie definitely fits here, with her extremely broad and extensive knowledge of technology.
Yellow is the Strong Guy/Tank: the team's hard hard hitter, the one who can take the most damage, and generally stereotyped as super loyal, and Hunk checks all of these boxes (his name even fits!)
Generally next in the list would be the Heart: the emotional center of the team, the one who keeps everyone together, the one always ready to give a pep talk, the one who makes sure no one is left behind. As the pattern follows, you would think this would go to blue (and so Lance), but it isn't. Allura is actually the one to fill this role.
Then where does that leave Lance? He clearly struggles with this question himself, saying he feels like a 7th wheel, like he's just an extra person. He struggles to find his worth on the team, surrounded by amazing people. This internal conflict over self-worth is what makes him so relatable and lovable to the audience, along with his humor. And here, I think is where we find his purpose.
Lance fills the role that I am going to call the diplomat in this variation of the five man band trope. He is the one that makes the team feel like it's not just the most amazing, perfect people out there that we could never compare ourselves to. He feels down-to-earth (pun intended), he's someone a lot of people can see themselves in. He's an extrovert who's great with people and longs for companionship. He's funny, and can make light of a situation, but when he needs to be serious, he is. Overall, Lance is a people person.
And to think of this in context of the show's universe, I think that position is quite important. Imagine if Voltron was real, and was what was protecting you and your home planet. Voltron is a great weapon, but it's larger than life. Even if it's protecting you, you'd probably still be at least a little scared of it, especially if the people piloting it were the best of the best. That's really intimidating. To have one of the pilots be someone you can see yourself in, it makes it feel more real, and less like it's something beyond your comprehension.
It's also important to have the diplomat for, well, diplomatic purposes. To charm those you want to make an alliance with, to mediate arguments, and to have someone the people of each civilization love. This part of the job is why Allura is able to pilot blue in the absence of Shiro.
However, she doesn't really fill Blue's role, and niether do any of the others. Allura is great, but she suffers from the same thing Shiro does in this regard, they're just too amazing. They're paragons of themselves. Allura is one of the last of the mythical Alteans, and she can perform the practically lost art of Altean Alchemy. Shiro is a perfect leader, he's handsome, and that makes him a little intimidating. Hunk is great in the way that he's loyal, but he's more focused on the people he cares about than anything else, and also just doesn't have the charisma that Lance has. Pidge is not the easiest to talk to, and you can find yourself getting lost in all the science lingo she throws around, and her long tangents that don't quite have all the context you need to understand them. Coran is a little eccentric, and that in itself can be off-putting at first. And Keith speaks for himself, our little asocial emo boy. (To be clear, this is not to bash the other characters, it's just to show why they don't really fill the role.)
All in all, Voltron is a deviant of the five man band trope, not the poster perfect example it seems to be. Furthermore, Lance is far from the worthless misfit he sees himself as, he is the true charmer of the team.
