That's what we do (although I audit the internal audit as I am independent from the process - it's managed by a colleague). The questions are: do we do it, does it cover the complete standard, are non-conformances handled correctly and closed off and are the auditors suitably trained?
It works for us.
I agree with BrunnyJim, even though you said you're not BRC, the BRC system could work for you. The internal audit for internal audits is just to make sure that your internal audits are covering everything they need to cover, and that corrective actions are implemented (and in a timely fashion). Also, it is very important as BrunnyJim says, that internal auditors be trained.
I think you could argue that the document you want to create can serve as an internal audit; being that the person doing the audit is independent of the process, and that person is checking the schedule and verifying corrective actions are in place. It seems to depend on the auditor and what they feel a valid assessment would be. Or you can just create an internal audit document the same as you would for other items in the Quality and Food Safety System, and have this person do an overall assessment of your internal audits.
We, like BrunnyJim, have an internal auditor do an overall assessment of the internal audit program. I keep up with closing out the internal audits and making sure corrective actions (including Root Cause Analysis, Proposed Action Plan, and Timescale) are implemented in a timely manner.