I'm not sure Mourinho showed the importance of it, he just chose to play strong teams in it. It was far easier for him, like Pep in recent years, to play stronger sides in it when they had/have squads big enough to play strong sides, without it impacting the sides they can put out in the League.
I said this last season, when we won it, and many times before, the League Cup is pointless now. For all but a tiny number of clubs it's nothing more than a distraction these days and the only reason we still have it, is because it's a big revenue driver for the EFL. It's not just PL sides that field weakened sides in it either, the vast majority of EFL sides do too. As part of the talks between the EFL and PL regarding the distribution of money down the pyramid, they are already looking at ways which will only further devalue the competition. It's been reported that due to more European football being on the cards, PL sides that qualify for Europe may not enter the League Cup in future or, as a compromise, they will field u23 sides.