I have no doubt that the team that get's knocked out of Europe first be it the Champions or Europa League has a huge advantage on those who aren't in it. Clearly Liverpool and Tottenham aren't challenging for the league, but any of the English teams could possibly end up in the EL.
Look at Liverpool 13/14, although they had an incredible forward line, no chance would they have been playing as well, or consistently had they had matches every 3-4 days rather than 6-7. I clearly remember saying at the time that they would have had to rotate more and that's where Brenda's tactical inefficiency would show, nor would Suarez be as fit/rested as he was every week to drag his team out of the gutter after they'd conceded 3 goals.
Although they still had the FA Cup to focus on, that run only lasted till early March iirc, whilst City were knocked out in the CL by Barca in the first round, but also had the COC until the end of February as well as making up the matches they had missed until then. Chelsea went to the same round in the FA Cup as Liverpool, but then had four difficult games against PSG and Atletico. Not to mention the FA's ridiculous insistence on not moving Premier League games to give their teams better chances of going through in Europe like the Spanish FA do for their teams. At one point we had Sunderland on a Saturday evening, Atletico away on Wednesday night, Liverpool away Sunday lunch time before Atletico on Tuesday night.
Would be nice to see the FA do that in the future, after all the moved one for a bloody EDL march, but I can't see the FA having that much common sense so doubt it will happen.