Do "to be" and "to have" work differently from each other with "each"?

Notes for context: I am a native BrEng speaker. I have read "Each of these is" vs. "each of these are", How does "each" change "are" to "is"?, and What should I use between "triple" vs. "all"? and I understand the general concept of "Each of them is" and how "each" always takes the singular.

Reading my daughter's bedtime story this evening, I came across the phrase "They each have a bag of equipment". It made me wonder why the following seems naturally (to me at least) to be true:

(Correct) Each of them has an X
(Correct) They each have an X
(Correct) Each of them is a Y
(Incorrect?) They each are a Y

I would never say (to use the examples from one of the posts I linked) "I have three pens. They each are green.", but "I have three pens. They each have a lid." is fine.

Is "to have" different from "to be" when it comes to using it with "each"?

Solution 1:

Janus Bahs Jacquet answered this perfectly in a comment but didn't respond to my request to re-post the comment as an answer, so I'm re-posting it here as a CW answer.

When used adverbially like this, each means ‘per [noun]’, which makes sense with have (distributive possession), but not with be (distributive being?). You can see this even more clearly if you move it to the end of the sentence: “The kids have three apples each” vs. “*The kids are boys each”.