Here's 4 types of legitimacy I can think of:
#1 All three characters are legitimate under the RAW. So how can any one be more legitimate than another? Simple. They aren't.
#2 A character played up from 1st to 10th is going to have a different build than a character made from scratch. They are both legit, but perhaps one style is more open to min-maxing depending your presumptions. Here legitimacy is based on simulationist expectations.
#3 The 1st character is more legitimate because it was actually "Earned" as you say in your title. What was earned? Everything that isn't part of the RAW.
As an analogy, the 1st character is like a 12th grader in High School. They have lived and learned all along, have friends, teachers, an integrated history, 12 years of trial and error studying, knowledge of the world around them, expectations about their school and life, ongoing stories and goals - a real bubbling ball of life. Now take this character back into to their 10th grade year, port them over to a High School in a different country and they are a fish out of water. No deep understanding of their surroundings, no support system, no integration other than their own identity. But they do have that - an identity. Not to mention a good idea of how the next 2 years of their life might play out (10th not 12th level, right?) So they are not so bad off. They have 10 previous years of learning to fall back upon.
To continue the analogy, characters 2 & 3 are like kindergartners with the physical and mental power of 10th graders. Everything is new. They have no experience. But they are starting off with the cool abilities of a 10th grader. Unfortunately, there is no back history on how they work. Or how to best apply them. Not knowing how the world around them works is a given.
#4 This last legitimacy is a confusion carried over from previous editions from 20-30 years ago. It actually doesn't have to do with "character legitimacy" at all, but was often thought to be the same. When it was tradition in the RPG community for ALL new characters to start at 1st level, a high level character was thought to be "Earned". Of course, having one wasn't a sure sign of player expertise, because you could go to any Con and find a kid with a 22nd level necromancer with the hand of Vecna. What was obvious to the DMs of the time was not obvious to the kid. That being, Character level =/= to player expertise. But it was far more common for the two to actually be equivalent. As character experience was more closely tied to a player's accomplishments in game (and All characters started at 1st, right? So it took some effort to get to high levels).
This legitimacy doesn't really carry over to the example you are using. The DM is choosing to begin at 10th level. The notion of players being qualified to run a 10th level character went out of vogue long ago. Now the only qualifications are knowing the rules. This is why some old-school DM's always start campaigns at 1st. I don't necessarily always feel the need. But I respect the reasons for their preference.