This is going to sound harsh, but please read it as offered in good faith. Also, the obvious caveat is that I don't know you or your director so these are generalizations not specifics.
>> My interview is dedicated to assessing candidates' facility and depth with this specialty.
Technical ability is one part of hiring. It's an important part sure, but as an employer I'm looking far beyond "ability to do the task." You note that he's specifically checking to see if the person is a "good fit".
When down in the weeds, it can be hard to understand the different boxes a candidate ticks. For example, if most of our team leader and junior manager appointments come from internal promotions, then I'm looking for potential in those spaces.
Your director may be looking for people with ambition. Or he may consider previous employment patterns (for us, we like long-term employees so candidates who job hob every couple years are usually a cause for concern.) Each company will have different criteria.
You don't go into detail here about your hiring experience, or your awareness of company goals beyond this specialization, or indeed if your earlier candidates were accepted or rejected.
>> I'm sensing what seems like some amount of distrust from the director about me and my team's capability to hire good candidates for this specialty
Trust is an interesting quality. We naturally feel we shoild be trusted by default, we naturally distrust others until trust is validated.
Any new director starts from a trust of zero. Your actions either build, or erode, that trust. Since the consequences of a bad hire fall on his shoulders not yours, he is wise to at least interview your approvals.
Your reaction to a veto, or even a pass, will help build trust. After he sees the candidate have a conversation with him. Try and glean what he liked, what he didn't like, and so on. There's a reason he's a director, and you are not.
Each interaction you have with him is an opportunity to learn. He is balancing variables above your pay grade, learning what those variables are, and how to factor them into your decision making is what builds trust.
Lastly, I'll point out that trust is not transferable. Your old director may have had complete faith in you. That counts for nothing. You have to build trust with this director from scratch.
Trust flows both ways. The more you trust your new director, and adapt to his requirements the faster the trust is built. The faster you adapt to him, the easier it is for him to understand your commitment. The more you dig into the new mindset (ideally by communicating directly with him) the more you adopt that mindset the easier it is to build trust.
As a new "boss" at any level, the worst situation is employees still hankering after the old boss, resistant to change, unwilling to move forward. The quicker you let go of the old baggage the faster your new trusts can be built.
You need to decide who's train you are on. And then get on. Or get off.