A name change is certainly warranted, but they seem to have missed the main reason it's warranted, namely: Your country's name starts with the letters Cz. Hello? It's almost as bad as starting it with Vq. (Note: the Cz is pronounced like ch, and the ch is pronounced like k.) If you want us to pronounce your name right, spell it Checkia. If you spell it Czechia we're gonna pronounce it Kizzeecheea. Also, do they have to put accent marks on all their vowels? It takes forever to write a single sentence, which is why their greatest literary work, Metamorphosis, is a novella.
Also, is it really that important to have Check as part of the name? When Burma and Rhodesia and British Honduras changed names, they went with Myanmar, Zimbabwe and Belize. Not Bzurmia Rzodia, and Bzshhndrs.
Here, of the top of my head, are a few better possibilities for the new name:
I'm sure you can come up with even better names. Which is why I'm starting the Twitter hashtag #SuggestedNewNamesForCzechRepublic.