Felix Felicis doesn't coerce anyone, and the only thing the potion is shown to manipulate is it's drinker.
Take the very first thing that happens - a jealousy spat from Lavender.
He pulled the Invisibility Cloak over his head and set off down the stairs, Ron and Hermione hurrying along behind him. At the foot of the stairs, Harry slid through the open door.
“What were you doing up there with her?” shrieked Lavender Brown, staring right through Harry at Ron and Hermione emerging together from the boys’ dormitories.
This isn't pulling on anyone's strings. Lavender was already jealous, this is just confirmation of what she already feels. She wouldn't have reacted like that for all 3 coming out, Ron and Hermionie are only following Harry because he was vague and evasive, and for some reason threw on the Invisibility Cloak before he even left the room. The only person's strings pulled here were Harry's, who did pretty much what was needful to set up this exact scenario - lucky for him.
Later on, with Slughorn, there is still no evidence of anyone being forced, or there wouldn't be a need for moments like this:
There was a long, long silence, but Felix Felicis told Harry not to break it, to wait.
Nothing is forcing Slughorn to act the way he does, drunk, morose, and full of regret. Nothing is pushing his reactions through this evening except, to begin with, his own greedy nature, and later, his own guilty conscience. Harry, with Felix's guidance, simply arranged a situation where he'd get plenty inebriated to lower his guard, and then with that lowered guard, honest enough to listen to Harry and come to his own decision.
Is this playing fair? No, and Harry had already tried this fairly. Is this manipulative? Oh yes, beyond doubt. Coercion suggests force, and Slughorn is not forced in any way here. The only person pulled to this result is Harry, by his own desire and the potion.