I am going to regret wading into this.
I am quite OK to say Czechia; then again, I have been exposed to languages that are quite OK to say Czechia (Tschechei, Tchéquie, Τσεχία, Ĉeĥio). So why the anomaly in English?
It could be an endogenous reason—because Czechia doesn’t work for English speakers; or it could be an exogenous reason—because someone told English speakers to use Czech Republic instead.
Is there something awkward in English about Czechia? Can’t be that the root is monosyllabic: we’re fine with Serbia and Bosnia and India. Can’t be the final -/k/ before –ia: Slovakia is OK.
Is it that Czech and Czech-oslovakia is familiar, and Czechia is unfamiliar? A minor factor, possibly, but I find it hard to believe that it was decisive. We didn’t freak out with Abkhaz ~ Abkhazia, after all.
So I’ll assume it was exogenous reasons. Someone, at the critical time of the Velvet Divorce, told English speakers that they should use Czech Republic; and Czechia wasn’t put forward as an alternative. Without exposure to the alternative Czechia, people went with Czech Republic.
Actually, no they didn’t, because that’s a damn fool thing to call a country. Noone says they went on holidays to the Commonwealth of Australia, or that they like listening to Republic of Korea-Pop. Without being given the option of Czechia, they started calling the country Czech.
Now, why were English-speakers not given the option? Presumably because some shmuck started getting all hot under the collar about how Čechy is not Česko, and we can’t have the world language conflating Čechy with Česko, and Moravo-Silesians are people too.
That hypothesis leads to the following questions:
1. Why did only English get subjected to that kind of edict?
2. Why did anyone in Czechia assume that anyone outside Czechia cares about the difference between Čechy and Česko?
3. Why did anyone in Czechia not realise that the small number of people who are both native speakers of English and care about the difference between Čechy and Česko, already have a word for Čechy—Bohemian.
4. Why did anyone in Czechia think they could get non-Czechs to understand that Czechia only refers to Bohemia, but Czech refers to Bohemia + Moravia + Silesia?
English does not have a committee running it, as Zeibura S. Kathau says in his answer. It does however have linguistic conventions and regularities. By making people say Czech Republic, the aforementioned shmuck was flouting the linguistic conventions and regularities of English. For that, they deserve to been bastinado’d.
Except that Fate has held an even better vengeance for that shmuck—and as collateral damage, all of said shmuck’s compatriots.
Fate has rewarded said shmuck with a generation of English speakers, saying that they went on holidays to Czech.