Just because there isn't a standard UTF-8 symbol (yet) doesn't mean ₿ is the correct answer either. 
₿ is widely known to refer to Bitcoin, not sats. 

You're, at this point in time, more prone to find someone that knows the symbol depicted below to mean sats, than ₿ - That's the current reality, that's how things evolved, for better or worse.
I think it's one of those "pick your battles" situations - Given how entrenched it is at this point, it's worth really asking yourself what has more worth - Fighting against this particular current, or putting energy into things that improve the current UX landscape.

Well-known sats symbol, as used in Snort.social:
http://blossom.proxymana.net/4c207969da6cc64f27284f4f4bcf26fa9acce050dff922bbc4e8950acf5c73a5.png