(Thank you Sindatur.)
Here's the problem.
The job of the companion is to ask the Doctor to explain what is happening so that the audience isn't confused.
The dumber the the companion, the more questions they ask, the more informed the audience is.
In this regard, Jo is a far superior companion to Liz.
However, Liz like perhaps Ian (The first male companion in 1963) is the Doctor's equal and should feel short-changed to be so marginalized below their station as an assistant rather than a associate.
Doctor Shaw didn't want to be just and merely a "companion" which is why she left.
U.N.I.T.'s Chielf Scientific Advisor would have got a reasonably impressive pay packet each week. The Chief Scientific Advisor's assistant less so. Did Liz get a pay cut when the Doctor stole her job (Which may be why the impressive academic might have really left.) and did she get paid the same as Jo did eventually get paid... Also that entire sexism thing, where women get paid a % of cash for doing the same job a man does, how much more money would they have paid the Doctor to do Liz's job, when he most probably wasn't even cashing his cheques?