Two things here:
First, I always like to think "Student" is for most classes the stage that happened just before you started your adventuring career; if for no other reason than to explain the jump in abilities between a commoner and a 1st-level character. Maybe 1-4 could be called something like "Line Worker" or "Neophyte".
"Zero levels" or "Novice"/"Apprentice" content should absolutely be a thing, yes. I have strongly advocated for such things since the beginning of the D&D Next playtest: there should be supportive, positive,
present in the PHB rules for creating "novice" characters who are still learning the ropes and may not even have all their "baseline" abilities yet (e.g. the possibility of needing to
acquire some of your background benefits.)
From there, you'd have "Journeyman," where you're formally initiated into your skills. You may not have learned every standard skill of your craft, but you're competent enough to ask for day-wages in it (which is the root of the word "journeyman," coming via French from the Latin
diurnus.) Then you'd advance to "Expert," having learned all the standard skills to a high degree of competence; "Master," someone who
could teach others the craft if desired; and "Grandmaster," someone who could teach
other masters. If each of these were 5 levels, that would get you to 20 (Journeyman 1-5, Expert 6-10, Master 11-15, Grandmaster 16-20.) 5e doesn't cover beyond 20, but further useful terms would be Legend[ary] and Epic, naturally.
Second, every edition - and 5e's no exception - has had a fairly clear "sweet spot" level range where things just plain worked; and for no edition yet has that sweet spot extended above about 14th level.
I mean, I'd argue 4e actually did go beyond that, but the things people found trouble with were different, and in fact quite fixable with an evolutionary, rather than revolutionary, update. But that's a discussion for a different thread.
Thing is, how can a designer do that without wrecking the sweet spot range where things already work well? 3e tried it by upping monster power commensurate with PC power, leading to something of a treadmill effect; that didn't work. 4e tried it by flattening the curve between levels such that the 1-30 power difference in 4e was less than the 1-20 difference in 3e, and for some reason that didn't work. 5e - well, I'm not quite sure what 5e did but it still seems not to have worked. And 0e-1e-2e didn't really bother trying; the DMs were pretty much on thieir own after about name level (9th-10th for most).
The only thing I can think of, but it wouldn't be popular, would be to dial down the power level overall - mostly by reducing the power gain at each level - such that a "new" 20th-level character would have about the overall power of what we now see as a 10th or 12th. Then, design for 1-20 as before and thus make the designed range and the playable range much more similar.
Well...what you just described IS what 4e did. If three levels in 4e is (approximately) equal to 2 levels in 5e, then a level 21 character in 4e is equivalent to a level 14 character in 5e. "13 is the new 20" is pretty close to what you're talking about here. Note that I don't say this because I think you are
wrong. I say it because I think you are
right, just conflating "4e had problems" (which it
absolutely did, both external and internal, both induced from outside and completely self-inflicted) with "because 4e did it, it must not have worked."
Spell levels 7, 8, and (especially) 9 are where the wheels truly come off. Where reality has to use metaphorical
safe words in its relationship with full spellcasters. The kinds of things a 9th level spellcaster can do--stopping time, creating pocket planes, literally enforcing their wishes on reality--are stuff that even legit actual superheroes struggle to do, if they even get the chance. 4e decided those things should belong in Epic tier. I think that decision was correct.