I lived in England for 7 years. The fans are irrelevant. No manager is playing squad players in a CL team to rest them for a PL game unless it's 6th game of the group stage and they're already guaranteed top. In the KO stages of CL, they rest their players during the PL games for it. At the start of the season, when you submit your 20 man squad or whatever for CL, it's the best of your squad.
The truth of the matter is, if an English team draws Barca, Real, Madrid, Juve, PSG, they have virtually zero chance of winning unless they are City. That's the reason the last 5 CL semis look that way, not because they're "not trying as hard".