This is just a thought, but I would approach the matter from an ellipsis-gone-wrong angle: It sounds unnatural when two phrasal verbs try to share what they have in common with two adverb particles. Consider the following:
I turned the TV on.
I turned on the TV.
I turned the TV on and off.
? I turned on and off the TV.
The problem with the last one is that ...
The phrasal verb construction in English stems from its origin as a germanic language. Take a look at this list for example under separable particles. For example the to give has a totally different meaning than to give up, so in German does geben distinguish itself completely from aufgeben.
In my opinion language belongs to the people who speak it more ...
The strange thing is that the order V Adv DO is idiomatic only when the DO (direct object) is heavy or the Adv is a single particle.
Admittedly, as pointed out in another answer, people do sometimes write phrases such as "turn on or off the tap". Yes, it's used, but, if the DO is light, people are far more likely to put it next to the verb. We are ...
Heavy (long and complicated) dependents are more likely to appear later in a clause. Hence,
We can turn the TV on or off.
It was the Conservative Government's idea that the way to control the
situation and deal with the problem was to turn on and off the tap of
consumer demand that was at the root of their stop-go policy. (British Parliament; House ...
"Check with" is verifying or consultating something/someone, before doing/saying something.
"Check in with" is more of a nicety; letting someone know that you are (aware that they are) still there. Or, in your example, following up on something.
Cambridge Dictionary makes this sense of check in clear:
check in [contact]:
to contact someone by making a phone call, short visit, etc, usually
in order to make sure there are no problems or to tell them that there
are no problems:
My son checks in regularly with me when he's travelling.
M-W goes straight to the purpose of making contact, which is ...
check in on (someone or something)
To actively monitor the security or safety of a person or thing.
Before I meet you at the mall, I need to check in on my grandmother to make sure she's feeling OK.
See also: check, on Farlex
To "check in on" someone means to visit, call, or write to them to
find out how they're doing:
When someone new joins the ...
In your context, you are dealing with two quite separate verbs:
To check transitive1, = to ascertain if something is correct.
To check in (phrasal verb) intransitive = to arrive for the purpose of contact. ("in" is an adverb.)
To introduce their complement, intransitive verbs need a preposition, which has a noun phrase as its object:
I checked in(...