From memory (they've been my 2nd club since my brother was somewhat tangentially involved as an acedmy player in the mid 90s), what was always claimed (including back in 1999 when it was pointed out how crass the iconography was) was that the Devon & Cornwall clubs typically called their 1st XV "the chiefs" as in, the bosses, through maritime traditions (claims of Celtic chieftains are pretty recent ret-con.s).
In 1999, when Exeter turned fully professional, they changed colours, started building a new ground, separated the 1st XV from the rest of the club, and rebranded to the Chiefs to keep that tradition alive. They chose iconography most easily associated with "Chief" - despite it being talked about as being crass at the time. Ultimately though, it was a little team from a little sport, half a globe away from the offended minority, so it was thought that no-one would particularly care.
Rugby has grown since then, Exeter have grown since then, and Exeter now appear on American TV screens every week. Society has also grown since then, and what was crass and a little insensitive 23 years ago, is unacceptably racist now - though that part could, and was predicted 23 years ago.
The problem has never been with the name (despite some of the straw-man arguments) but the iconography