There’s no speedy way to gain your staff engineers’ trust.
Staff engineers have been around the block. They’ve ‘seen’ things. They’ve ‘lived’ through things. They’ve observed successful changes within an engineering organization and, of course, they’ve observed unsuccessful changes.
They’ve solved, and are continuing to solve, the hard problems. Sometimes hard problems are in the form of strategic investments based on business needs. Other times it’s identifying gaps in the technology that enables people to do things to make the product work.
As an engineering manager, you’re unlikely to be successful in your role without the support of your staff or most senior engineers.