Root words are supposed to be the smallest base word, with all the suffixes and prefixes removed. The definition of the root word is supposed to be maintained, and is the base for any new words created by adding suffixes and prefixes.
The suffix -ist = a person with a particular set of beliefs or way of behaving.
The root word for theist is theos (god). Attaching the suffix -ist defines a person. Literally, a person who believes “god”, is true.
The prefix a- = without, no, or not.
The root word for “atheist” is still stated to be the Greek word “atheos” (no/not/without god). Maintaining the root word, and attaching the suffix -ist defines a person. Literally, a person who believes “no/not/without god”, is true.
When we turn to talk about “theism” and “atheism”, then we have swapped out the -ist suffix, for an -ism suffix, while maintaining the root words.
Well, okay then. What does a-theist mean? It literally means “not a god believer”. To attach an a- prefix, the new root word, for this new definition of “atheist”, should actually be “theist”. And, instead of swapping out the suffix to discuss “atheism”, we’re actually discussing a something with a different root word. The root for “a-theism” would be “theism”.
Because personhood is attached to the -ist suffix, and an a-theist isn’t an -ist, this word has not defined a person. A rock is “not a god believer”. It has also not created a philosophy that people are following. Unlike athe(os)-ism, which described a philosophy, a belief that we are without gods, a-theism, “no/not/without” “god belief” describes every non-sentient thing in existence, and most of the sentient ones, as well.
To make a-theist describe a person, you have to rip the personhood out of the “ist” suffix and transfer it to the “a” prefix, so that “a”= “someone without” and “theist” = “god belief”. Wait a minute, though, “theism” = “god belief”. Now the “theist” in a-theist and the “theism” in a-theism mean exactly the same thing, and “a” sometimes includes personhood and sometimes doesn’t. Either that, or “atheist” and “atheism” mean the same thing.
Broad definition atheist supporters argue for the word to be constructed like no other similar word. Many compare it to the word “amoral”. The problem there is that there are such things as amoral-ists and amoral-ism. “Amorlaist” means “someone who believes we are without morals”. “Amoralist” doesn’t mean”not a moralist”.