Actually they did and prefered to say BRD rather than West Deutschland if they had to make the difference with the DDR.
On a daily basis, it was of course Deutschland but in a more formal and administrative context, BRD was commonly used and accepted.

Bundesrepublik Deutschland was (and still is) the official title of the country. I agree, though, that most laypeople would have just said "Deutschland" or "Westdeutschland".
As West German, it was definitely difficult to say "Deutsche demokratische Republik" (the official name for east Germany) without chuckling over the "democratic" part, or saying it in a mock-Honnecker (east German general secretary) accent.

In formal German, people would have said BRD in West Germany in the 1980s.