You do leave a paper trail when cashing a check or even depositing a check.
Cashing or depositing checks does leave a paper trail.
The bank also keeps records of any checks you cashed or deposited and the deposits get legally reported to the IRS.
When you cash a check at your bank the cashed check will be listed on your bank account and the bank will keep records of that cashed check for awhile.
So yes cashing the check does leave a paper trail and whenever you cash enough checks they can also get reported to the IRS.
Cashing checks also leaves an electronic trail and the only way to not have an electronic trail or leave a paper trail when it comes to money is to use cash when possible.
Even when you use debit cards, credit cards etc those all leave electronic trails which is like a paper trail.