Should people be allowed to disallow certain admins to review their (un)ban requests?

[RIC0H]

We in staff team handle the cases unbiased and we are never biased for either parties.

A recent player report would beg to differ on that one.
I'm not going to link it as everyone should already be aware of it.

I don't see it working that disallowing certain mods/admins posting on courthouse reports that involve them or their friends, they are as much part of the staff team as others.

I agree, if they have input on the case and are involved in it, then they should be allowed to post.
But not decide the outcome.

cases are always reviewed and all staff members look into them UNBIASED.

Sorry, but that is simply not true (regardless of how much you believe it is).
There will always be some bias present in such cases.

The way I try to resolve it is as follows:
If one of my staff members reviewed an appeal that they issued a ban for, they would be demoted.
If one of my staff members reviewed a report that they're involved in or are friends with the person being reported, they would be demoted.
If one of my staff members contacted the case handler and tempted to alter the outcome of the case, they would be demoted.

The above may sound harsh, but it's effective in keeping the bias out of cases.