Google alert failed to warn people of Turkey earthquake

why is a private US company supposed to take care of the communication of such events to Turkish citizens?

Because Google has established itself as one half of a de-facto worldwide cellphone duopoly. It vets and serves the vast majority of the apps that runs on its platform from its app store, it puts a large majority of the world's population - including Turkish citizens - under surveillance without their consent, and all the push notifications from all the apps running all over the world are sent through Google's servers.

When you set yourself up to corner a country's citizens, limit their choices and exploit their private data to that extent, you owe a thing or two to that country in return at the very least.

Most people still refer to it as Turkey, so it's still Turkey (although it's not wrong to write/say Türkiye).

Different languages have different names for countries. For example Germany vs Deutschland, or Olanda vs Nederland. Turkey can want whatever it wants, it's hard to dictate a country's name in other languages.