Thing is: Trevor already has his personal reason.
Trevor still is his own person with his own goals. He has been banished from the living with other people because he's seen as some freak of nature with vampire powers (yes, the Belmonts have been specifically banished because their powers are confused with vampires'). Earning the right to walk amongst the people again and renewing his actual perceived humanity is reason WAY beyond enough. Dracula is to blame for their banishment -- he's a vampire, he has displayed powers that compound the fear people have of the Belmonts.
He doesn't need another personal reason from a personal vendetta of someone who lived 400-odd years ago. The Belmonts have already spent this time hunting other vampires and beasts of darkness. For me, it's much more believable to think that they have outgrown the vendetta after not finding Mathias for so long, and just accepted that they hold power to banish evil, and that this power should be used responsibly. That NOT using this power to protect humanity incur on not only irresponsibility, but in a case of "I helped nobody, now that I need help nobody is here to help me". On a Batman analogy: Joe Chill (Mathias) is the symbol of evil that spurred Batman (Belmonts) on their hunting of evil. He's not in to catch Joe Chill in specific, but what he represents -- Joe Chill will be caught eventually because he's a criminal, but that's a footnote on the bigger "hunt all evil" page Joe Chill started.
To the Belmonts starting at Trevor, Dracula is a total stranger. Yeah yeah "ancestor vendetta" but the ancestor in question is long gone and 400 years is a LONG-ASS time with no contact with the actual offender (Dracula) to give it too much importance.
Another very, very important detail is: How would they even recognize Dracula as Mathias? Just being a powerful vampire isn't enough, dude. By Dracula they have already seen vampires abound. They don't know what Mathias looked like, they don't know what name he chose for himself, and even if they had a painting of Mathias survive all the way from Leon up until Trevor (damn unlikely), Dracula's face by Trevor's time is far from being recognizable (as we've seen it depicted, at least).
Like, I would be much more convinced if they had actually witnessed Dracula doing Draculy things all through these 400 years, having their mind refreshed on who the guy at a constant pace. But 400 years with no word of the man? For all they cared he could have died already by their own or some other hunter's hands. He changed identities, remember? It's the point of a disguise. Nobody will recognize you given enough time.
Not everything needs to be "ancestor vendetta" levels of personal. Every Belmont has their own personal stake with Dracula. Be it "Dracula kidnapped my girlfriend" or "Dracula mind-controlled my son".
The Belmonts knowing who he is makes the confrontation more intimate. And that is always better.
Eh. Not sold on that. You can't be believably intimate with a guy from 400 years ago. You may even know there is a vendetta, but in truth you don't KNOW this guy. You haven't quite built your perception of him because all you know are tales from ages past. It's like meeting Hitler and thinking you have an intimate confrontation with him because you read too many WWII books. You don't KNOW the guy -- you just understand the basic concept that he is really evil.
It may be better when revealing Dracula's identity has any importance. What importance does telling the people who Dracula is has at all? What importance does knowing he was some guy on the crusades 400 years ago has? None. It's not like the Church could do anything against him (they are already trying their hardest, and failing miserably were not for the Belmonts). Knowing who Dracula is would go on more or less like this:
Belmont: I HAVE COME TO SLAY THEE, O MATHIAS, BETRAYER OF BARON LEON BELMONT!
Dracula: Oh cool, 'tis been ages since I have last heard this name.
Belmont: I KNOW THY NAME, O INFERNAL DEMON OF LEGEND!
Dracula: Nice. What will you do with it? Try to scavenge some occult knowledge from my long dead heritage which I have obviously destroyed on this centuries-long timespan because I'm a genius tactician and don't plan on having my alchemy used against me? Tell Aunt May?
Belmont: I WILL REVEAL THAT YE HAVE LIVED FOR LONGER THAN THE COMMON MAN!
Dracula: What, wasn't that cat already out of the bag when the whole vampire stuff became obvious?
I'm joking here, of course. But I'm trying to express that I see no advantage of knowing who Dracula is beyond "I know who Dracula is". I don't think an ancient vampire warlock would even bat an eye to someone knowing his name. Like, remember Bison? Same concept here: Betraying Leon, for Dracula, living long as he has and seeing the shit he saw through all this time, was a Tuesday.
Also, another important thing: This "I know who you are, DRACULA!" stuff cheapened Mirror of Fate immensely for me. Dracula has been already waging war on humanity for long, but all Trevor thinks is "U MUH FATHER". This "ancestor vendetta" trope is an easy way to write antagonism between two characters without actually having to develop a reason for the character itself to have a personal vendetta to hate the other party. It CAN be done very right -- I'm not saying it can't -- but the stars must be aligned. One such star is seeing the guy your father hates doing shit, then the son still sees the same guy, then his son, then his son... Keeps the vendetta flame alive.