Well, the thing is, your approach is (more or less) "get 5 incremental advances immediately, and 5 more at the end of the first session."
Whereas trying to work backwards from your approach to my (proposed, untested, spit-ball-idea) approach is...well, impossible. There are no other incremental states, just "half a 1st level character" and "all of a 1st level character."
Well, yeah.
The point of the original post is, the level 0 already exists, and works well.
I understand these are sudden jumps in power, once when generating the level 0 character, and once again when leveling up. But this is the way 5e is. For most people that need a level 0, this works well. For me, I expect this level 0 to typically comprise one gaming session with one main challenge (whether social, exploration, or minor combat), unless players get into something interesting on their own.
That said. A more incremental approach is possible. All of the background features can organize into feats. Then characters gain a feat after each session, until they fill out all of the background, and afterward level up to level 1.
It is slightly confusing to refer to "feats", since a level 0 "feat" is equivalent to half of a level 4 "feat". For clarity, the feats here are level 0 feats, and not level 4.
Level 0 Base
• Species Traits
• Speed: 30
• Age: typically 16−19 years old
• Languages: Common and two languages of your choice that you and your DM agree on
• Standard Array
• Hit Points: 5 + Constitution
• AC: 10 + Dexterity
Level 0 Feats
After overcoming a main challenge, choose one of the following level 0 feats. After completing all five feats, overcoming your next challenge gains level 1 in a class of your choice.
Feat: score +1. Repeatable: you can choose this feat three times, but only upto twice for a same score.
Feat: two skills, one tool
Feat: Background Feat
With the above in mind, level 0 might stretch out incrementally for about six sessions before finally leveling up to level 1.