I have no problems with surgeons or university professors administering their departments, albeit my experience is that many of them are terrible administrators. I have a problem with marginally qualified people being put on jobs where their relative incompetence creates risk.
I think that may not be a problem in many industries (like retail, catering, amusement, etc.) but it could be a serious problem in other (esp. health care, transportation, and mfg, as well as certain other services). So assuming as you do that job specialization breeds hierarchy, this would pose a serious problem - we should either compromise safety for the sake of equality or leave some "bastions of inequality" intact for the sake of safety.
My preferred approach is to recognize that there will always be inequalities among people, including inequalities in desirability, power, wealth, skill etc. - the point is not to eliminate them, but to recognize their existence and implement social control mechanisms that prevent them from totally poisoning of social life as they do it now in the US and many other countries.
Again, my experience form the "other side" of the iron curtain tells me that wage scales were rather flat, director/worker wage ratio was, on average, in the vicinity of 3:1, and the main wage difference was between the sexes and industries (i.e. some skilled manual workers earned considerably more than "techno-managerial class" depending on the sex and industry). What did that remarkable, by the US standards, pay equality mean for social inequality? Several things.
One was contempt for higher education among the working class who often boasted to earn more than engineers. Government efforts to encourage working class to pursue higher education (extra points on entrance examinations) were met with stiff resistance from the intelligentsia, but otherwise did little to attract sons of workers and peasants, who had an opportunity to earn more as welders or machinists.
Another was that other things than money became signifiers of social status - some of them as silly as the possession of certain consumer products (blue jeans, for example).
Third was the growth of nepotism and grey economy - people trying to get things they needed (not just consumer goods, but also work to be done) via informal connections.
Finally, there was a group of people that did not want to be empowered - they preferred living marginal existence in squalid slums, drinking vodka every day, supporting themselves by extorting money from "their" women (or by selling sex) - and there was nothing anyone could do to change their ways. And their social background/upbringing had very little to do with what they did - I know people ending up as bums and drunks while their brother and sisters had college degrees.
Wojtek