The reason Mbappe left PSG was basically to win Champions League title. He did his best to win it at PSG but seems luck was never on the luck of PSG because I don't see anything that stopped them from winning it for several years that they assembled the best players in the world. There was a time they had Messi, Neymar, Mbappe and Di Maria in the same team yet they could not win it. Maybe Mbappe have realized that it may not be easy for PSG to win it, the reason he left for a team that have better records and chances of winning it. His decision to join Madrid have nothing to do with bad performance at PSG where he was almost worshipped.
Actually, we don't know what his main reason moving to Real Madrid. But there are 3 possible reasons:
1. Real Madrid is his dream team since he was a kid.
2. Real Madrid offers the chance to win UCL trophy.
3. Playing with Real Madrid will enhance the opportunity to be the best player in the world.
You're right. PSG never has a luck to win UCL trophy. Even with Messi-Neymar-Mbappe, PSG still couldn't win UCL trophy. I think it is about the quality of the coach, too. If PSG hired Guardiola at that time, I think PSG may already win UCL trophy. But the coach of PSG at that time, has no good experience in UCL competition. IMO, to win UCL trophy, a team should be managed by a coach with good experience in UCL.