Minimizing trust should always be a goal of a security system. If you can minimize trust without harming usability, compatibility, capability, security, cost, etc... you should do it.
When we talk about trust we often mean different things:
* In cryptography and security by "trust" we mean a party or subsystems that if they fail or are compromised then the system may experience a failure. I need to trust that my local city is not putting lead in the drinking water. If someone could design plumping that removed lead from water and cost the same to install as regular pipes than cities should install those pipes to reduce the costs of a trust failure.
* In other settings when we talk about trust we are often talking about trust-worthiness. My local city is trustworthy so I can drink the tap water without fear of lead poisoning.
As a society we should both increase trustworthiness and reduce trust assumptions. Doing both of these will increase societal trust. I trust my city isn't putting lead in the drinking water because they are trustworthy but also because some independent agency tests the drinking water for lead. To build societal trust, verify.