I would really like to know how this question is not off topic:
While Slovakia is an acceptable short name for the Slovak Republic, it seems that Czechia should not to be used for the Czech Republic. Why?
The question basically sets aside the linguistic topic (the analogy from Slovakia) and then asks why. I mean, the question eliminates the major linguistic reason as it is presented.... there is no other possible 'why' or 'should' about it. The only thing that can be said is to offer another opinion or to say that a country can call itself what it wants.
How is this not opinion based, as far as an English word goes? Even 70% of Czechs said they didn't like it.
If the answer is largely based on history and geography then one has to draw the line somewhere and say this is a social sciences question.
The only possible tie-in to English is that Czechia follows the pattern of Slovakia. The question eliminates this from the get go. Anyway is not necessarily an established pattern. The accepted answer talks about historical English, which is also officially off topic per the help section.