Yes, whatever you want to optimise is a good way to phrase this. However, it'd not just what you want to optimise, you will get side on effects. Even when things seem reasonable, longer term things come out. I don't have a good solution for this. But training people to target one (or even a few) things is incredibly difficult.
I also question the trust aspect. Part of this (which can also be gamed of course) is the level of trust you have that a team is doing their best to work towards some goal. More metrics indicate lower trust in some ways.
I think specifically this (metrics vs judgement), gaming happens in both, but metrics lead to much more wild problems if gamed, because they have a much harder "you said do this more so I did" backstop than "well this seemed to be what you wanted".
Thinking about it a bit more, if you make the argument that human judgement here is itself a compound metric of many different inputs, this semi resolves itself. But we still have the problem that the metric has no backing that can be handed around other than "seems good to me", which changes for many people.