Actually, I believe you have made a good observation on this.
I have discovered that "Review" means different things to different people. for some, it means just making sure a date, signature, and no blank spaces and such are on the form. Others, they check procedure, proper training level, legible values, etc.
I check all the above, plus applicable standards and procedures, assure printed data aligns with as found/as left statements from vendor, uncertainty/standards used information, etc.
This is just my opinion, and we all know what they're worth.
I think it's worth having a procedure for standards. That one procedure addresses who do you send it to (approved vendors), how it should be packaged (very important for standard instrumentation), marking instrument out for calibration, checking equipment in, reviewing documentation, assessing out of tolerance condition, and returning to active service. The procedure can be very short, but at least you have a guidance document where everyone is doing it the same.
Then, you could (if you chose) have a checklist, router, etc. for the instrument when it is returned.
Whether you have a procedure or checklist is totally up to you, but assure that everyone is doing it consistently, and that the calibration certificates are being verified for the information essential to assess fitness for use, and potential non-conformity impact.