The prince was suffering from profound loneliness? That's the excuse for his behavior? She was too in her first life? That's her excuse? Really? Participating in the Prince's inner circle and being his royal cockhouse was out of loneliness?
Did the author forget the beginning where she said everyone loved her as a cute child so she became selfish lookist who did whatever she wanted in her first life? Couldn't think of a better reason for her to still feel uneasy like the state secrets that got her hanged?
We don't really know why he was a jerk, but I feel like the fact he was popping up at tea parties as a girl and being bullied was an implication that MC bullied him too in her first life.
Anyways, she probably would have walked to her death regardless. There's no way she wouldn't have friction with him because of her established selfish personality.
Scheming? Knowledge? She was just something the prince bounced on his waist that happened to learn state secrets and accidentally leak them. She didn't start studying seriously until her second life.
Who knows why she doesn't have at least a guard or two and some servants while at the orphanage though.