Again, I gotta believe there was a rule about only submitting one song per film.
I think you are right, but I have not been able to find official confirmation yet. I have several books on the Oscars. I paged through every chapter of Inside Oscar from 1934 through 1980 -- there is a chapter and a section on rule changes for every year, but I couldn't find any mention relating to changes in the nominating process for Best Song. From 1934 through 1979 no movie ever received more than one nomination for best song, then suddenly in 1980 the movie Fame got nominations for 2 different songs.
I am sure that over the years I read somewhere that there was either a hard and fast rule or perhaps just a tradition that a movie was allowed to submit just one song for consideration. Apparently that must have changed in 1980 or in the years just before it. I recall a huge uproar in '64 and '65 when no song from either A Hard Days Night or Help! got an Oscar nom. The issue was raised again in '78 when the blockbuster hit film musical Grease got a single song nom for "Hopelessly Devoted to You" while other original tunes like the title song and "You're the One That I Want" were ignored over some pretty unmemorable songs from mediocre films.
I'll keep looking through other books and will post if I find something concrete.