For my money, everyone could always just consider things like this-- a PC can choose and has a subclass at Level 1 even if they do not get a game mechanic for it to use until Level 3. You can just roleplay it.
You can be a Drunken Master even if the Monk abilities you are using at levels 1 and 2 are not specific to the Drunken Master and are just part of the standard Monk. It's absolutely fine. Especially when you consider the fact that the game mechanics you do get at Level 3 only look like Drunken Master abilities because of the fluff that those abilities are written down to have and surrounded by.
What are the game mechanics you get at Level 3?
- Proficiency in Performance
- Proficiency in Brewer's Tools
- Use Disengage as a bonus action
- Gain 10 feet in walking speed for a round
If anyone was to look at these abilities in and of themselves and were asked "What class would these subclass abilities apply to, and what do you think the fluff of that subclass would be?" I think few people would have any real idea. They'd certainly try and guess something... but no one would actually be able to ping it first try. Because these game mechanics are all pretty generic all things considered. And as a result, there's no reason why anyone should think "Oh, I HAVE to be able to do these four things at Level 1 in order for my Drunken Master Monk to FEEL like a Drunken Master Monk." Especially considering the fact that a Rogue character with the right Background can pretty much have or duplicate all four of these features at character creation. Are they thus Drunken Masters by default? Of course not! Because the mechanics do not denote what the subclass is... it's how you characterize your PC and roleplay it that gets its subclass across.
Here are another set of Level 3 subclass game mechanics:
- 10' aura that causes 2 fire damage to all enemies within aura
- 10' aura that causes one enemy to take 1d6 lightning on failed saving throw
- 10' aura that gives all allies 2 Temp Hit Points
Do any of these abilities scream out "Oh, these have to be for X class obviously!" I would say no. None of these three features tell us what class they should be, they are again three relatively generic game mechanics that do not in any way give any indication what class they should be for, or the fluff or narrative of the potential subclass concept for that class. Which means it isn't important that these be given out at 1st level because they do not do anything to denote your subclass.
What actually gives a player the indication of what the subclass would be is the fluff that surrounds those game mechanics. It is the fluff that we read that tells us that these three abilities are all for the Storm Herald Barbarian. But the thing is... we can just apply that fluff to our characters at 1st level by how we roleplay them. We can roleplay our barbarian as being a herald to oncoming storms, even if we won't receive an individual game mechanic for it until Level 3. And that's perfectly fine, especially considering that the game mechanic we get gives no real indication of what it's meant to symbolize anyway.
If you are a fey-themed Glamour Bard... you can be a fey-themed glamour bard even if you don't get your first unique game mechanic until level 3. You can be a paladin who has taken an oath of conquest at 1st level even though you won't get a mechanic for it until 3rd. If you feel like an assassin then you can play as an assassin right from the start. It's all about how you see your character and play your character, and not what goofy-ass mechanics you have written down on your character sheet.
Of course, that's just my opinion... I could be wrong.
