A bill recently signed into law and coming into effect on 9/1/21 is Texas House Bill 723. It’s a reasonable modification of the Texas Health and Safety Code. When a modification is made to a death certificate that is not currently in pending status, then the person making the modification is required to notify the deceased person’s next of kin that the death certificate has been modified.
This means that no one can go back after the certificate has been fully filed and modify it to add or remove anything with it without notifying the next of kin. So there is no option for someone to quietly update the reason a person died later on.
See the final text of the bill: https://capitol.texas.gov/tlodocs/87R/billtext/html/HB00723F.htm