Mac Mini frequently loses connections (both Wi-Fi and Bluetooth)

Is it likely that replacing the Wi-Fi/Bluetooth adapter would resolve the problem?

Yes.

It's highly likely that the issue is the Airport card because it is a combo Bluetooth and Wi-Fi adapter. Given that you are having issues with both functions, it points to the adapter being at fault.

You've already done a significant amount of diagnostics and ruled out other machines don't exhibit the symptoms connecting to the same network. You can further verify that your Bluetooth peripherals are not the issue by connecting them (keyboard and trackpad) to the MacBook Air. You could also run AHT - Apple Hardware Test. It may detect a problem with the Airport card.