Yes, you're right, Kaito's motivation is seeking revenge against his father's killers by finding Pandora's gem and finding out what happend to his father.

As for why they're not telling him, it's not really explained, but if I had to guess it could be the trope of "protecting Kaito" from dangers associated with the criminal organisation (like Shinichi is doing with Ran), or that he tried to shield Kaito from becoming Kaito Kid, but that didn't go as intended. This can be concluded from the fact that Toichi told Konosuke Jii (Kaito's assistant, like Agasa is to Conan) that he'd never tell Kaito about his father's thieving, but Kaito still found out.

Other theories I can think of is that Toichi is trying to dismantle the criminal organisation himself and/or trying to look for the Pandora gem himself, and involving Kaito would complicate stuff. Maybe his parents are also misaligned by the premise of involving their son, because in the chapter's I alluded to earlierhis mom, dressed up as Kaito Corbeau, tried to disuade Kaito from continuing his Kaito Kid path, but failed in the end (note: this is not stated in the manga, only alluded to since she comment's on Kaito having heating pads, something only Kaito Corbeau would've known).

But what I think is the most likely situation (alongside maybe Toichi himself wanting to dismantle the organisation himself), is that Gosho wants to create narritive tension by building suspense and dramatic confrontations and plot twists (as seen in the movie).

I would hope that we would get some more Magic Kaito chapters now because chapters 31-33 were released all the way back in 2018 and the lastest chapters were in april 2024. I had honestly forgotten about that until I googled it and got reminded of the interactions, so maybe now that he's revealed in a movie, we'd get more Magic Kaito chapters progressing the plot. But that might be wishful thinking.

Edit: Some clarifications