Yes, a common and accepted genderless pronoun.

It has been commonly used and accepted in the case of unknown or unspecified gender, though it is rapidly falling out of favour for that case. In part because I think it's quite hard to say that it's truly 'genderless', it's embedding a 'default' into the language which perhaps is undesirable.

When has 'he' ever been used in a genderless context?

Apparently it started in the 18th century, so there's a lot of inertia to overcome to get people to change.

But she says it was from the 18th century onwards that people started using male pronouns when describing someone of a non-specific gender in writing and this marks the time when opinions on what pronouns should be used started to change.

"You might have a sentence like 'if a student comes to see the teacher, he must bring his homework', where he is supposed to refer generically to males and females," explains Dr Emma Moore.

https://www.bbc.com/news/newsbeat-49754930

I have been learning Italian. Don’t slam me for this if it’s wrong I am a beginner. But there are a number of contexts in Italian to which lei (she) is used. One for polite and respectful use (such as a general in the army, regardless of gender) or if the person’s gender is unknown.

Just thought I’d throw that out there.

Very formal/polite way to refer to people yes. But in Italian we also have “voi” and its use can be traced back centuries before “lei” started to be used.

Welcome to the English language.

It's technically correct to use He when the gender is unknown.

Provided by a sibling dead comment, see 2nd definition:

https://www.merriam-webster.com/dictionary/he

https://en.m.wikipedia.org/wiki/To_Be_a_Pilgrim is the first example that springs to mind.

Dominique (a non-gendered name) goes to school everyday. They take the bus.

Is this a proper English sentence?

(question from a non native speaker)

As a native english speaker I consider that valid.

The possible confusion is the fact that “they” could be singular or plural. AKA “singular they” [1].

Your second sentence could be referring to some other group of people who are not Dominique who instead of going to school take the bus.

[1]: https://en.m.wikipedia.org/wiki/Singular_they

"They" is a plural pronoun. Obviously.