The answer is that it the shortest and least recognizable word to describe a country in existence, as it is a simple percussive sound of only one syllable.
Chad is the only other country that is as inarticulate as Czech, you may say that Guam, Greece, France, Wales, Spain, are also one syllable country names, but they are actually diphthongs.
It isn't obligatory in every country to say Czech Republic, and in every language, it is a question of accepted norms.
The Czechs don't actually say Czech Republic, they commonly say "I come from Czech" when they speak English. "in Czech we like beer". "in Czech we have strong civil liberties".
It is an accident due to the hasty division of Czechoslovakia. At the time of decision, no one could think of an alternative better suited to the English language.
Czech is an annoying sound to pronounce and an un-descriptive name for a beautiful country, It is the same as having a country called Psip, pshek, Kst.
It is inconvenient to use on it's own and doesn't sound like English, and interrupts a phrase with a percussive sound which cannot represent a country by nature of it's sonority.
It is partly a question of politics, but mostly of language and semantics.