When you say conservatives support Meritocracy do you mean like how Ivanka and Jared got jobs with the White House? When people criticize the word "Meritocracy" they are not criticizing the concept but the usage.
I don't agree - there is a subset of people that criticize meritocracy as a concept because they believe the idea of 'merit' is inherently racist and classist - if you start at a different level, it becomes more difficult for you to accrue 'merit' and so that needs to be balanced and taken into account. Some people believe this is much more important than hiring or promoting on 'merit'
The people who write about these things always say they don't like when the word meritocracy is being used to hide bias.
Here's another: https://practicaltheorist.wordpress.com/2012/11/14/the-myth-...
Here's a critique stating that the terms aren't defined enough to be said to be 'just'
http://assets.press.princeton.edu/chapters/s6818.pdf
The last one is the most fascinating, I really recommend you read it if you're interested in these things. Amartya Sen is brilliant.
Though I suppose it's easier to dismiss complaints about bias when we pretend those complainers just hate our meritocracy.
I'm not sure how to parse your last sentence. Could you clarify what you mean?