One may brew coffee in shabbos using a pour-over cone.
There is no problem of borer. This is like the gemara in shabbos that is brought in SH"A which allows one to pour water over wine sediment that is in a sieve and drink the wine flavored water that comes out. This is not borer because you aren't separating an existing mixture.
There is a shaila of bishul. Anyone who makes tea on shabbos (sfardim or kli shlishi like R Moshe) can definitely make coffee. I heard from Rabbi Feurst, although he doesn't allow tea he allows coffee in a kli shlishi because it's bishul achar tzli