Just for fun (and realistically because I was avoiding things I should really be doing) I went back and did the same quick analysis for barbarians. Once again human. Target AC 16 for average damage. Dear old "A" sucks, "B" is far more effective in and out of combat.
The conclusion is that "A" never catches up on damage no matter what they do, their AC sucks, they have far less HP. You could mitigate the AC a bit by getting breastplate (so long bare-chested warrior) but they'll always fall behind on HP, AC, long range combat, initiative, and skills as well.
Not to mention giving up on multi-classing for A.
Level 1
A: is going to be a pretty bad barbarian, but that's what he wants to play.
B: Not just a good barbarian but pretty good at other stuff as well
level 4. Note that they get primal knowledge, an extra proficiency so I took survival
A: Increases strength and dexterity to get an ever-so-slightly-better AC eventually
B: Increases dex and wisdom by 1 so they're better at some out-of-combat skills.
level 8.
A: Takes polearm master, increases dex. I'm assuming using the bonus action for polearm 4/5 turns.
B: Sticks with the greataxe and has a lot of options. For simplicity increasing Con by 2.
A: is going to be a pretty bad barbarian, but that's what he wants to play.
- Str: 15 Dex: 10 Con: 14 Wis: 9 Int: 7 Cha: 4.
- Attack: +4, DMG: 1d12+2(avg 4), HP: 13, AC 11.
- Initiative +0, Perception +1, Stealth +2
- Should probably buy some armor, use a shield and give up on the greataxe. But we're comparing apples to apples here.
B: Not just a good barbarian but pretty good at other stuff as well
- Str: 20 Dex: 15 Con: 16 Wis: 13 Int: 13 Cha: 9.
- Attack: +7, DMG: 1d12+5(avg 6), HP: 15, AC 15.
- Initiative +2, Perception +3, Stealth +4
level 4. Note that they get primal knowledge, an extra proficiency so I took survival
A: Increases strength and dexterity to get an ever-so-slightly-better AC eventually
- Str: 16 Dex: 11 Con: 14 Wis: 10 Int: 7 Cha: 4.
- Attack: +5, DMG: 1d12+3 (avg 4), HP: 37, AC 11.
- Initiative +0, Perception +2, Stealth +2, Survival +2
B: Increases dex and wisdom by 1 so they're better at some out-of-combat skills.
- Str: 20 Dex: 16 Con: 16 Wis: 13 Int: 13 Cha: 9.
- Attack: +7, DMG: 1d12+5 (avg 6), HP: 45, AC 16.
- Initiative +3, Perception +4, Stealth +5, Survival +4
level 8.
A: Takes polearm master, increases dex. I'm assuming using the bonus action for polearm 4/5 turns.
- Str: 16 Dex: 12 Con: 14 Wis: 9 Int: 7 Cha: 4.
- Attack: +6, DMG: 1d10+3 + 1d4+3 (avg 11), HP: 69, AC 12 (15 with breastplate).
- Initiative +1, Perception +2, Stealth +4, Survival +2
B: Sticks with the greataxe and has a lot of options. For simplicity increasing Con by 2.
- Str: 20 Dex: 16 Con: 18 Wis: 14 Int: 13 Cha: 9.
- Attack: +8, DMG: 1d12+5(avg 15), HP: 93, AC 17.
- Initiative +3, Perception +4, Stealth +6, Survival +4
The conclusion is that "A" never catches up on damage no matter what they do, their AC sucks, they have far less HP. You could mitigate the AC a bit by getting breastplate (so long bare-chested warrior) but they'll always fall behind on HP, AC, long range combat, initiative, and skills as well.
Not to mention giving up on multi-classing for A.

