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 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 
 Yea good points. From my bubble, my friends / target users have no idea what a sat is, but they all know what bitcoin is, so i'd rather send them the asset then send them a unit name and explain the asset. I still think its very very early when thinking in unit terms and most people just look at the dollar equivalent value of bitcoin they own. 
 I think there's also a perspective worth evaluating whereby trying to explain unknown stuff, we over-complicate in the process. 
In that regard, if we want to keep it as simple as possible, the explanation/comparison of Bitcoin to the Dollar and sats to the cents (or whichever fiat currency they are familiar with), is in that context an extremely easy one for a new user to grasp because it conjoins the familiar with the unknown. 
And maybe, that's all we really need. 

It just needs to be clear when a user is onboarded (Note, I don't think we need to SHOW the fiat value, just convey the comparison). 
Also worth noting as Bitcoin's value rises, sats may also be less and less attainable - And I think that's where your thinking comes in of keeping to a singular name which is understandable - Keep it simple, whilst allowing expansion. 

I think it'll be a wait and see game, because sats really did gain a lot of traction with the introduction of Lightning, and rightly so.  

And personally I think all this will come very very naturally - Up until relatively recently, the bitcoin unit was very attainable by the average person. In the present, not so much, and yet we're seeing more and more adoption - A string being pulled in seemingly opposite directions. It'll work out in the end.