I think this is a misconception. Or in a more positive way: Wishful thinking. All of the alliances provide a documented set of benefits. Things like F lounge access for OW Emeralds or extra baggage for *G.

And you only get those written down benefits, which are actually not that many.
Everything else (arrival lounges, upgrades, service etc) are still things which fall outside the scope of the alliance benefits so each player sets their own rules. Some airlines offers services to other elites which they don't have to give (UA waiving baggage fee for *S for example) while other try to engineer their way around having to give benefits (BA by having the Concorde Room, LH by not allowing A380 F awards etc).

There is one side of me which thinks a bit like you: Each alliance airline should treat any alliance elite like their own. But there is also one side of me which thinks that different benefits for different people is a good thing and helps with the development of the overall industry. And keeps FT going.