The king knows the true nature of the second prince. My prediction is that there is someone behind the second prince or the second prince who was responsible for the first prince's abduction. The second prince believes that he needs to be the king and is manipulating people to be viewed positively. I think that he believes that might be the only way he gets "attention" or "love"
The way the manga is portraying him so far, he is most likely to be an antagonistic with a redemption arc rather than an outright villian.
I find the king here a little detestable. Cliches of a romance fantasy and the obvious antagonist setting of the 2nd prince aside, I feel that what you pointed out as him (the 2nd prince) possibly being emotionally neglected is a fair deduction to make—which just pins this whole situation on the two adults who were supposed to have been there for the child as guidance and support.
The 1st prince disappeared while he was 3 if I recall what I read correctly. And considering that the 2nd prince was likely younger than that (barring any being born to the queen vs the mistress scenarios with succession issues) when his brother disappeared, it’s very unlikely he had anything to do with his brother’s misfortune.
Even if his personality and actions are now warped, honestly I can understand why he’s doing what he’s doing. Do I think it’s right? No, def not. But honestly if I grew up with parents (very likely) neglecting me to find a sibling I barely remember while enduring successor lessons and any bad adults that likely try to latch onto me in the parents’ absence, only to then later have all my hard work go down the drain with the arrival of the ‘true successor’ who btw needs rehab and therapy at bare minimum, and have my father tell me that this is the one who’s going to be king now… idk, maybe I’d at least consider stabbing him (the father) down his family jewels.
The mother (as shitty as she is to say that the ML is no longer their son) has a point. The first prince shouldn’t become king but if the second prince is lacking somewhere in character to be excluded from succession so suddenly, honestly the argument feels moot when weighed with the first prince’s current psyche. Like… at that point, the obvious you reap what you sow when it comes to upbringing aside, isn’t it better to find a successor outside the direct line and just collectively all have therapy as a family? (Though of course not often an available option for historical based fantasies… and also, for the sake of the cliche plot development)
Maybe that will be explained later but I really don't get why the king wants the ml to be prince at all cost. Why not make the other prince the king and give a title to ml like duke status or archduke. He doesn't even want that shit it only brings trouble