Conrad is loyal to the throne, while Alaina is loyal to the elves. So it's natural to want to remove Alaina. Because she will only support MC as long as he is on the side of the elves, otherwise she will become his enemy.
No, we can't say that "Conrad is loyal to the throne."
We have literally no basis for that.
We can't get inside his head and find out everything.
What makes you think he's not interested in the throne itself?
Again, if he finds proof of the real Regis's death, what basis do we have to say he won't use it to take the throne himself?
Here's an even more pressing question: what benefit does Elise gain from anyone finding this evidence?
It's rationally stupid to allow Conrad to have this information.
Yes, Alaina is loyal to Eliыe because he's a half-elf, and she has a vested interest in Elise using his power to benefit the elves.
But...
Alaina killed and manipulated countless people so that Alice would be the sole heir and take the throne.
What did Conrad do to put Elise on the throne? Nothing.
What is Conrad doing to keep Elise on the throne? Collecting evidence of his illegitimacy?
Are you really considering whose skull to smash with a goblet, choosing between the one who gave you everything and the one who's collecting evidence against you?
Are you sure you're thinking logically?