AMC's should earn their living the way any other business does - by charging their customer ( a lender) for the management service the AMC provides to the lender. The lender, if they want to use an AMC, should compensate the AMC the way a lender compensates other service providers like accountants or IT support.
But the AMC business is "special", in that it typically does not charge their lender customer, the AMC gets compensated as a pass through payment of a split from the appraisal fee the borrower paid the lender. That sets up an incentive for the AMC to seek the lower or lowest fee from the appraiser, because the AMC earns $ from the margin of the split. Which translates to it costs the lender nothing, as in zero $ to use an AMC ( thanks to the perk of bundled fee on the HUD ).
Name another business that operates this way. I can't name one. Most AMC's are not out to "get" appraisers, and might be run by some decent folks, but their business model means they are onboard with getting their profit from a fee margin. It sounds to me like you are making an assumption, ( outside of a few rare special assignments ) that the lender client reviews the bids and tells the AMC which appraiser to use.. It makes no sense, because the reason a bank client uses an AMC is to relieve them of the burden of appraisal management.