Technically two of the lords in Three Houses meet the criteria of "playable at first, but become unplayable and unrecruitable after certain points in the story." You play all three in the prologue, and when you choose to teach one of their Houses after the prologue, the other two "leave the party" permanently. They're only playable for one chapter (outside of DLC, anyway), but so is Orson.
Depending on how hard and fast the "become unplayable and unrecruitable" criterion is, you could count Dedue in the Blue Lions route too; he will only rejoin you in Azure Moon if you did his sidequest during White Clouds, and there is no way of knowing in advance that that is a prerequisite for him rejoining you.
By a similar token to Dedue, Wallace in the Blazing Blade will only show up in Eliwood's/Hector's tale if the Lords' levels are low enough in the mid-game to trigger the map in which he shows up; otherwise, he arguably "becomes unplayable and unrecruitable" after Lyn's tale too.