Our calender is not so much a calender, as a set of rules which determine every year whether it should be a leap year and whether cheshvan and kislev should be 29 or 30 days.
There is no rule that asara bteves cannot be on friday, and none of the other rules affect it either.
Yom kippur cannot be on friday because we don’t want two days in a row of absolutely no melacha, hence we say that rosh hashana cannot fall on Wednesday. The other fast days also cannot fall on friday, because the other rules make it impossible.
Asara bteves on friday happens fairly often. Much more often than Birchas hachama.