How could it have a profound influence if its got no control over free will? If its profound it should be able to change ones personality or take away free will, which it doesn't.
Free will doesn't have to be affected in order to influence someone's actions. What has to be affected is one's perception of what is going on around them. For instance (and here I'm using Spike only as an example, not as any kind of comparison to Angel) if Buffy perceives Spike as the Champion who saved Sunnydale and the world, the person who gave her the strength to carry on when all of her so called friends betrayed her, and the man in whose arms she wanted to be in what very well might have been her last night on earth, then she is almost certainly going to trust and depend on him in this new crisis.
If, on the other hand, she sees him as the person who. upon returning from the dead, never bothered to let her know that he was back, therefore abandoning her just the same way everyone else that she ever cared about has, who rejected her declarations of love, and who, just coincidentally shows up at the critical moment with the vital information that may help her to quell this latest apocalypse, who obviously has some sort of unknown agenda, she is going to see Spike as simply a resource to be used but never truly depended upon.
By manipulating Buffy's perceptions of the things and people around her, by planting just the slightest doubt as to someone's loyalty, by creating just a slight distrust of the methods and agenda of a potential ally, Twilight doesn't actually change Buffy's, or Angel's, sense of right or wrong or their desire to serve and protect mankind. It simply alters their perceptions of what make an action or person good or bad.
The same can almost certainly be said for how Twilight manipulated Angel into the belief that leading an army and killing hundreds of Slayers was actually the "right" thing to do in order to save Buffy, or, for that matter, that saving Buffy at the cost of everything else in the universe was the best choice
😱 under the circumstances.
Twilight couldn't make Buffy or Angel kill a friend, but it could alter their perception of that friend to the point that they were now seen as actually being enemies, at which time killing them would seem to be the proper action to take.