I wonder if rounding the displayed skill to the nearest 100 or 200 would keep people calmer about it. You could still use it to balance teams manually as necessary, but you wouldn't see fluctuation game to game so much. It's definitely not designed as a progression system to motivate people, which would probably make different decisions.
It might be worth representing the public face of the number as some sort of ranking:
2,200 to 2,399 = Silver, 1 Star
4,800 to 4,999 = Shadow, 4 Star
600 to 799 = Recruit, 3 Star
etc.
That way everyone will be obsessing over their "league" or "ranking" or whatever instead of their number. I don't think it would fundamentally change anything.
I'm more keen on getting the server browser ping display "fixed" for higher latency players rather than seeing eveyrthing as red. I guess the equivalent would be entering a rookie server only to see that everyone has 1 bronze star because everyone is under a set hive score value.
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The default ranking of the server browser should mostly take care of that for you, even if the UI makes it hard to see.