The whole idea behind Lindsey's view on the world is that he doesn't want to end up like his father. He says it in this very episode, how disappointed he was in his father, and he doesn't want to be like that. That's what drives him, that ambition, the ability for him to be able to say 'I did it, I made myself to be a better man than you ever could'. Lindsey isn't a bad guy, and we can see that by the way he helps those children, he's able to put his ambition aside to worry about them, why he helps the people who are being used for organ harvesting in 'Dead End' and it's the same when he steps up to help Angel in NFA. When he needs to, he can put someone else's needs before his own.
But that ambition and that drive for money to make sure he doesn't end up back in that Southern town, living hand-to-mouth, like the rest of his family, comes back and overwhelms him, and it's for this reason that he accepts Holland's offer at the end of 'Blind Date', because he's scared he'll end up like his daddy and that fear will always, always drive Lindsey. And it's for that reason that Angel - hypocritically - decides that Lindsey is too much of a risk, especially after NFA when W&H will be without a CEO or any sort of earthly representation. Angel was too worried about Lindsey's ambition taking over again and Lindsey ending up in charge and doing some serious harm to the world. Which is why he decided that Lindsey had to be killed. It's one of the only things I hate Angel for.
