Is major kufr automatically also shirk?

While "kufr" and "shirk" may both be translated with the shorthand "disbelief" by some translators, there is a distinction between the two in Islam. Kufr is mostly about rejecting Islam, while shirk is about accepting any deities instead of/besides Allah. However, this distinction is not entirely clear to me, for the following reason: when someone rejects Islam, they will have some mental model of how the world functions, and necessarily that mental model will not be grounded in a belief in Allah as described by Islam (otherwise they would not be rejecting Islam). For example, Islam teaches that everything in creation was created by Allah and is sustained by him and only by him. Someone who does not believe in Allah will have some competing idea, such as things exist by themselves, some other godhead sustains them, maybe a monotheistic god created them but now put a power in them by which they sustain themselves, etc. All of these beliefs would be shirk if I'm not mistaken. Due to this, does kufr therefore imply shirk, in the sense of: every kafir is automatically also a mushrik?