Maybe the key of a song is totally based on the melody and not the harmony?
even with no melody, the chord sequence you are describing is certainly in the key of C. .....It's as if we're trying to figure out whether a donut with sprinkles is still a donut, even though the sprinkles are not made of donut. indeed it is still a donut, especially considering the decorative materials in question are in fact very standard classic materials, used in the very standard classic way. and no one in history has ever dared to call such a donut a non-donut