The reason they will not let you do this is simple - as a TL you need an ETL/STL to sign off on a counseling. (a "write up") They do not agree that this situation warrants a counseling, and therefore they do not want to sign off.
You are thinking of this as them trying to take away your authority, but the problem is that you have zero authority to issue a counseling. Only an ETL or above has that authority. When a TL does a counseling they are simply reporting the offense. TLs do *not* have the authority to approve a counseling in any way, shape, or form.
So basically in regards to counseling this is the way it works:
ETLs - Issue the counseling (i.e. the official paperwork, or actually signing it if they delegate the writing to you), approve the counseling, and decide what consequences the counseling entails.
TLs - Report the issue to an ETL for them to consider issuing a counseling, deliver the counseling verbally should the ETLs decide that is best
As a TL, you have done your job by reporting the issue. The ETL involved has decided not to approve a counseling.
You seem to think that you have the authority to approve/issue the counseling by yourself and the ETLs are taking this authority away from you. In fact, you do not technically have this authority. No TL does. Trying to take this to a higher level is only going to get you in to trouble, and possibly even a write up your self for insubordination.
And yes, as I say this as a fellow TL. Drop the matter before your ETLs see you as needing corrective action.
One more thing, just an FYI in case you are thinking of trying to do this - a TL issuing a counseling without an ETL signing off on it is a termable offense for a TL. Whatever you do, don't go behind your ETLs back and try to do a counseling without approval. You can and will get termed for it. The reason Target has this rule is - TL's do not have the training for all HR laws like your ETLHR does. All it would take is a TL doing a corrective action for the wrong reasons, (i.e. something in violation of your states/federal labor laws that the TL was not aware of) and the next thing you know Target has a multi-million dollar lawsuit.
A "coaching" is a totally different thing, and you can do this by your self. The problem is, it has zero power behind it.
Most TMs think that when we as TLs do a "coaching" without an ETL it is some dire thing. The fact is, a coaching is *not* an official corrective action that has any official negative consequences for a TM. I can sit around and "coach" a TM every day for months - that TM will still have a job, and will not suffer any negative consequences until when and if an ETL decides it should become a counseling. A coaching is simply generating a paper trail for your ETLs to use as "evidence" against a TM to justify a counseling. But a coaching by its self does not impact a TMs job.