Is there any subset of the German language similar to "simple English"? By "simple English" I mean a subset of English easily readable by those whose mother tongue is some other language.
Yes. In German, it's called "einfache Sprache" (simple language). There's also a variant called "leichte Sprache" (easy language) that goes even further in simplifying the language for accessibility purposes.
Some public offices as well as media outlets have offerings in simple language to make their content more approachable. For example the public media outlet "Deutschlandfunk" offers news in einfache Sprache. Another example is a collection of translations into einfache Sprache of common applications forms for things like child benefit (Kindergeld) or residency.