How do I deal with a player wanting to do an activity that takes weeks when the party doesn’t want to?

I’m currently running a campaign with 5 players at level 3, one of which is a Wizard Dragonborn who’s entire character is devoted to becoming a real dragon by the end of the campaign through some kind of magic or holy gift or something.

As part of this, the player has decided that he absolutely MUST have a Pseudodragon and has decided that next session, as the players have finally returned to town after leaving at level 1, he is going to spend a few days resting before setting off alone to explore the forests of the nearby area to find a Pseudodragon.

I explained to him that on foot, sweeping the whole forest systematically (in a frontier part of the world where the majority of the land is forested) will take literally weeks for his character to do, as he has no spells that can assist him except find familiar (which he could use to sweep the air with a hawk).

When I explained that, this would involve him as a player turning up to the session (online) and contributing essentially nothing for extended periods of time over the course of several sessions (my players have decided they will be leaving town soon and our sessions have very little time between them in the world) he decided that he was fine with that.

I really don’t think that he will be and I’d hate to lose one of my players because in two session’s time he decides that he is really bored and doesn’t want to keep playing but on the other hand I really don’t feel like it’s fair for the other players just to give him what he wants immediately because I’m scared to lose a player.

I know he has said that he is fine with it and I’ve explained the downsides to doing what he is planning to do.

As the DM, is there a better way that I can damage control this? I don’t know if I’m making the right choices by not giving the player what they want but I just cannot see a reasonable way that a player could quickly find a rare animal in hundreds of square miles of forest.