Pros and cons of the rivalry/friendship system

It still allows you to disagree with them more openly and more often than an approval based romance would, though.

On one hand, the rivalry doesn't mean hatred. On the other hand, this can mean hatred, and while this system works smoothly if Hawke not in romance with the rival companion. The romance is more complicated than that. It depends on the characters, and what Hawke think about the "love". S/He needs a real partner, or just someone who good in the bed.

Isabela: The rivalry romance working well, I think, if, Isabela doesn't get bored from her serious partner, later...
Merrill: She's a blood mage, and she will not give up this. She working on this mirror, what perhaps is meaningless for Hawke. If Hawke considers to idiot Merrill's whole obsession about her people's history, then I think, the romance is not really healthy. Perhaps there is some chance to change, if Hawke help her to find a new goal (work for the city elves).
Fenris: The rivalry can working well, if Hawke not supports the slavery. There are some idiot answers in this version, but no one of them hurts Fenris. In rivalry Hawke only wants to convince him, that s/he or his/her sister also deserves freedom, just as anyone else. Friendship or rivalry with him only depend on Hawke's temper, because Hawke can be honest radical pro-mage without any compromise, and still support him. This is clear, if at the Act3 Fenris finally able to continuing their love-relationship.
Anders: The rivalry romance with him a mere sexual attraction, without any ideological/spiritual interest. "I hate your politics, I hate your 'demon', but I like your sexy tortured gaze and nice ass." But Anders loves Hawke. The rivalry romance will crush him.

Te system have many flaws, but very enjoyable and believable, just as Hawke's somehow "carton-like", but surprisingly real character.

Would have worked just fine in DA:O & Inquisition. Would have improved Inquisition ten-fold if they would have focused fewer more interesting romances, allowed rivalry and made them more plot-important and more talkative and dropped off all the fluff and way, way, way too many characters with little to nothing to contribute. Should have been Dorian, Cullen, Josephine and Cassandra, all in-party, and cut out completely the pointless "filler" characters like Sera, Blackwall, Vivienne, Iron Bull, Varric and Leliana.

Ah. I think the feeling is mutual. I suggest we acknowledge the difference of opinions and leave it at that.

For me, the system in DA2 is the best of the DA series and even the ME series. I think the companions are as well. I have enjoyed my time in DA2 more than in the other games.

I like it in theory. But in practice I felt locked into going "all or nothing" with their approval or disapproval in dialogue or who to bring with me when making choices, so I wouldn't lose them at certain key moments. If they didn't get lukewarm feelings from both agreeing and disagreeing on things (which is normal in relationships)... then maybe I'd enjoy it more.  It's also weird that someone who rivaled you 100% is more likely to stick around than someone who is "meh" about you.

I get that, but the lukewarm feelings simply because they don't agree or disagree 100% on everything always struck me as odd. They've been spending time with each other for what? 7 years? And just because they have a dynamic relationship of different agreements and disagreements it's lukewarm? I didn't like it.

My main confusion is with Anders, because Justice and belief in his mage cause go hand in hand. Only in rivalry, can Hawke tell him joining with Justice was a bad idea. Friendship encourages him, even if you just want Hawke to be pro-mage liberation.

Another example was with Isabela. I don't remember what I did wrong that one time, but Hawke talked to her plenty and took her around but she still got the hell out of Dodge at the end of Act 2 because they were still too neutral.

Anyway, I felt it was too tedious of a system with Friendship and Rivalry working against each other, so I found myself doing a lot of meta-gaming to plan their relationships before talking or going around Kirkwall doing stuff. I didn't care for it myself. Others apparently do, and that's fine of course.

I think Anders case is logical: Hawke can't befriend with him, if consider him an abomination. Of course in friendship, Justice's okay. Justice his most sensitive point, I think a friend try to help him to keep his sanity, instead of endangering his life because of an irreversible process. I really like that the friendship confirms his faith and strengthen him even against Hawke, if s/he betray him at the finish. Bethany, for example, is his friend. Probably she never would capable of doing such a thing, what Anders did with Justice, but she understands him, and wants to help him. Some people likes rivalry with him, because they want to "fix" him, with keep away Justice from him. But this is a dangerous nonsense: Justice already inside him, and he's not a broken car to fix... 
Of course, the friendship/rivalry system isn't perfect, but somehow explainable and a lot of fun.

I know, what you want to say, but why Hawke wants to crush him, if believes, that he's right and feel, he's a friend? Because of Justice? After Karl and Ella's case, Hawke can express his/her feelings about Justice (negative or positive too), and still can stay in friendship with him. And as people said, the rivalry's open with him, even if Hawke supports the freedom of the mages.
I probably see, your problem is, that Justice doesn't take the control over Anders in friendship. But why he would do such a thing, if Hawke doesn't threat Anders?

Yes, it's unjust, that friendly Hawke doesn't see Justice and even Anders works with manifestos. Only thing can explain that: Anders doesn't have any significant problem with Justice in friendship. 

Many people do not like the system as you are, I just tried to explain some of the open questions, but you also have good points in this theme, because there are many flaws in the system. For example, back to Anders, WHY rival!Hawke accepts Anders' requestfor Elthina's distraction? Rival!Hawke 
1. doesn't want to free mages;
2. wants to free mages, but doesn't trust Justice, then doesn't trust Anders...

About that this topic becomes another Anders debate: the whole DA2 about the mage issues, so he's inevitable.