Because a draw is a draw, there is no need to specify how the game reach a draw.
Why no notation for stalemte?

Stalemate should be: - or -- (maybe even _, !#, or #!)
Unclear position should be: * (because it's on people's keyboards)

Because a draw is a draw, there is no need to specify how the game reach a draw.
Precisely.

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I like this question!
Checkmate has its own symbol even though a win is a win is a win. I don't see why stalemate shouldn't have a symbol too. Of course, coming up with one that is instantly recognizable could be challenging. Maybe the pound sign (hashtag sign for the post-millenials) with a circle crossing it out?
Like this:

Being a pedant here, but the OP graphic is incorrect in its depiction of castling. Castling notation should be written using the capital letter 'O', not zeroes.
If you look in old books, the zero is being used instead of the letter. I suspect this has changed because of the PGN format, in which the letter is mandatory, as it is easier to handle for parsing purposes.
Similarly, I was first familiar with the symbol × –sometimes ++– for checkmate. This could as well have changed with the rise of PGN.
As for the stalemate symbol, well, I’d like to see %. But I guess all this doesn’t really matter. I reckon no symbol should be specific to a language, though.
Stalemate should be: - or -- (maybe even _, !#, or #!)
Unclear position should be: * (because it's on people's keyboards)
While * is on the keyboard, it’s also the PGN symbol for game ongoing or terminated without a result. Wouldn’t be practical.

I like this question!
Checkmate has its own symbol even though a win is a win is a win. I don't see why stalemate shouldn't have a symbol too.
A draw by three-fold rep or a draw by insufficent material need their own symbols too?
Actually, it would make far more sense to me to have designations for losses both on time and by resignation since you can't look at the score to determine these causes.
I really think "!" and "?" should be replaced by and

👈😧😴❓😶😬👹😟💅😶❄👍👊😜😳♌✴♌🚭💇🚶💇💤💇👅🇹🇷🇪🇬⚓✈🇦🇷🇮🇷🚥🚇🚚🚊📔✒🔆📺💽🔊💽⌚📉📔📭💡📮🎅📩📅🎤📔📪🐫🐐🐠🐍🐵🐓🌿🌏🐖🐙🐖🌐🌗💐🐧🐔🐘㊗♿♌💹◾🔗🕗🕧🔜✖🍥🍔🍥🍧🍤🍧🎿🍞🍚🍈👒👘💛👙🎀💎💬👤
Here, alot of notations to chose from. Maybe there should also be notations for mate in one and queen capture.
There is no symbol for stalemate because resultwise it's a draw. Chesswise it is irrelevant because if you know chess you should be able to realize that it is a stalemate.
Also this is what comments are for. There is no further need for symbols.
Simply play through the moves, and when you reach the end, ah-HA! Stalemate!