Okay, as much as it's probably a waste of my time... let me try and explain it this way, as maybe this might make things clearer. I don't play with the whole "living world" idea that many people do. What I play and what is real is what happens at the table when it occurs at the table. That's it. In each and every campaign I run, only that which actually is presented in the story at the table is real and true. Nothing else. Anything that is in the Player's Handbook? Rules... abilities... races... classes... until they appear in the game in the story at the table, they don't exist. And it might never exist. Things in the Monster Manual? They don't exist until it shows up in the story in the game.
Does this run counter to a lot of other people's games? You betcha! No doubt! They have this whole campaign that plays out in their head, with thousands of events happening that are in no way connected to the players and their PCs at the table, and which the players never know or hear about.
Well, I don't have that desire or time to make all that crap up, nor pay attention to what all that stuff is and when it's happening in the background. It is not necessary. At least not for me. Instead, my game is simple-- if it occurs in the game, then its real. If it hasn't yet... then its not. It might eventually become real... for instance maybe the cockatrice makes an appearance in the game when a PC decides to do some research on monsters that turn people to stone and I introduce the idea of the cockatrice for this story. But until that happens... who's to say whether this setting, this story, in this game HAS the cockatrice? We just don't know. Or more to the point, no it doesn't until it eventually does. But we have no way of knowing until it occurs.
So the answer to the question of how can a 1st level wizard be a "great wizard"? Because in the game as we play it... if no other wizards have appeared in the narrative yet, then that wizard PC is the only wizard in the world up to that point, and thus that PC claiming to be a "great wizard" or "the greatest wizard known to man" is completely true. Because as far as the other characters in the story have seen up to that point... PCs, NPCs... that wizard PC is the only one that exists and thus ipso facto they are the best. "But what about wizards who can walk the planes"? Well, until one of them actually appears in the story-- there are no wizards that walk the planes. That's not a thing in this particular game. NOTHING is true or real until it appears in the game, at which time, THEN the fiction accepts and supports it.
And like the 25 year veteran of improvisation that I am... I 'Yes, And...' the crap out of everything. And if in that first session or two the player of the wizard PC declares they are in fact "the greatest wizard known to man"... even if the character is only 1st level... then I'm going to accept that declaration as true. Why? Because why not? I find interesting and compelling characterization F-ing cool, and I have no problem whatsoever moving the game as we go forward continuing to support that narrative choice AS TRUE. In THIS PARTICULAR GAME the player made a character choice, and I will support that choice by not negating it out of hand. Why would I? Why yuck their yum like that? Why would I ever do that to that player? Now sure... I will of course challenge that PC going forward-- you need drama to keep the story interesting and compelling. Other wizards will most probably show up in the story down the road and the player will need to defend their characterization if they wish to continue using it... but until those wizards actually appear, they don't exist, and nothing has yet negated what the player put forth.
And all of this is possible regardless of the level of the game, or the mechanics used. Story is what matters to me. Character is what matters to me. Events that actually occur are what matter to me. Any other stuff? Don't care. Not until it shows up and actually impacts the story and characters of what is happening at the table.
And that's it. If all of this still makes your eyes go crossed and your head swirl in confusion... then just accept that you play the game as far away from how I do as possible and that you will never understand it. And I'm okay with that.