Saying that Loghain has nothing to apologize for is way too far.
It's admirable that he's willing to face death or Wardenhood with dignity. However, let's not forget that he brought those fates on himself.
He does admit to that. "All of this can rightly be called my fault."
Or I guess he shouldn't have to apologize for: Allying with Howe (a known traitor, torturer and murderer); Selling elves to Tevinter (Immoral and Illegal); Hiring assassins to kill the Wardens; Framing the Wardens for something that they didn't do; Poisoning Eamon; Capturing and imprisoning a templar; or attempting to bypass the Landsmeet and seize power for himself. (even if Teagan didn't say something, the Civil War was clearly on its way regardless)
What's the point of an apology at this point? I mean, really? What would it mean? If he said, "Hey, yeah, sorry about Howe. That guy turned out to be a real jerk." Would it matter? Would your opinion of Loghain change? The way Loghain addresses his actions at the Landsmeet is something that I can admire. He's not making excuses or trying to weasel his way out of anything, and I'll be the first to agree that Loghain made mistakes, and several of them. Most of what he did was for the sake of political expediency and he doesn't shy away from stating almost exactly that. An apology would be meaningless and trite to me. Once Loghain decides something he fully commits to it, and while I know he has regrets, I would have been disappointed if he had said, "Sorry about that" if only because I wouldn't believe him. He did what he thought was best at the time, but with incredibly high stakes. He found out he was wrong, and it wasn't because of anything he did, but because he didn't fully understand the situation or the Warden's role in it.
Putting aside the clear fact that the Recruitment crisis is clearly contrived for forced drama (I've pointed this out before), there isn't much reason to recruit Loghain at the risk of losing Alistair.
I wish I could talk Alistair into staying around, but since I can't, I will readily give him up for a guy who's actually willing to do the job. Alistair shows himself time and time again to be unwilling to make hard decisions, and this case is no different. Alistair abandons his duty because he doesn't get his way. Now, this is all opinion-based, but just based on that I don't see a reason to keep Alistair and lose Loghain.
It's also interesting that folk often bring up Zevran and Sten in defense of recruiting Loghain. Why not hire Loghain if you're willing to hire an assassin and a professed foreign murderer?
Well, you're not hiring Zevran. He joins you willingly. And the opportunity to "hire" Loghain never presents itself. He's needed as a Warden, not as a mercenary.
I'd argue that the difference lies in their particular situations. Hiring Zevran is a clear risk that can backfire as TEWR has pointed out. However, he has no real vendetta against you and you've already proven that you can defeat him (in fact, you can outright kill Zevran after beating him). Sten makes no excuses for himself and holds to the life debt that you forge with him unless he utterly loses respect for you.
Loghain doesn't have a vendetta against you personally, but has a well-founded mistrust of the Wardens. I don't really care about Loghain's mistrust, because I just need to get him on my side.
With Loghain, you lose more than you gain from recruiting him. Why trade a trustworthy and battle-hardened templar for a paranoid and irrational old man? I'd sooner put up Zevran or Sten for the Joining than Loghain and I like Loghain's character.
Alistair actually proves that he can't be trusted to carry through on a sworn duty. The biggest decision he's ever made in his life is to become a Warden and he actively chooses to abandon that duty. So I'm going to have to disagree with you about him being trustworthy. While my Warden is not ready to go so far as to call Loghain trustworthy after the Landsmeet, he more than proves his worth.