"it is natural that curly braces can denote both a set and a sequence, depending on this implicit input."

?

I don't even know where to begin. Overloading symbols in mathematics occurs all over the place. There's nothing wrong with that. The difference between overloading a symbol and abusing it is whether there is an agreed upon definition/convention regarding its use and to what extent its use conforms to that definition/convention. What I'm saying in my original post is that the statement "{x_n} is a sequence" disagrees with the formal idea of what a sequence is and that most writers don't bother to explain their own notational use.

If you wish to re-define the curly braces to have a context-dependent meaning, knock yourself out. But, I would imagine that that practice would confuse a lot of people. Math is a human activity. It's not a programming language.