It's a pretty shitty colonial term with no real relevance today... but unless you come up with a better name for the province and get buy in from people (which will be fucking hard) it's the word.
That said, this article is way overblown... this isn't a public mandate but instead pulled from an internal style guide and, yea, not talking about British Columbians when speaking to indigenous groups is probably a good idea.
I think there was a proposal floated with the idea of a new name in Chinook, "Illaheechuck", but it failed to - as you say - get buy in from the public at large. I like the idea of a name in Chinook, though - what better bridge between native and settler than the language they mutually created?
Personally I would love it if we changed the name. It's too long and a pain in the ass. Plus it's weird coming from a place that's two different places merged together.