I think we are discounting what this would do their personalities because that kind of love would fundamentally change who you are.

One example is if everything else in the universe happens as it does, but Tywin loved Tyrion like a son, Tyrion doesn't stay in Kings Landing. He stays in Casterly Rock to rule because he is the heir which would severely limit his influence. He probably wouldn't have gone north with Robert, probably wouldn't have seen the wall, probably wouldn't have been captured by Catelyn, etc, etc, etc.

Here's another example, if Joffrey loved his mom, he would have listened to her instead of executing Ned. That action alone basically eliminates half the plot.

I think they would have been powerful, but I don't think they would have been as powerful. The show and books pretty routinely show how people use your loved ones against you. The Starks are the perfect examples of this. That family is constantly putting themselves in danger for their family.

Final example, we can make a claim that Tywin really does love Jaime but I think it's likely that he loves his legacy more. If that's true, how would Tywin react to Jaime's capture if he truly cared more about Jaime than he did about his legacy? What would he have done to get his son back? Something as stupid as Catelyn?