I'd admit to being one of those who doesn't care so much for the cups these days.
While I can't deny that nothing tops winning trophies, I think the sport has evolved in such a way that the PL and CL stand above all else for obvious reasons. And there are now a multitude of other factors at play. First and foremost there's fixture congestion, which is even worse due to the pandemic. Games come so thick and fast that the natural inclination among fans is to rest players for the cup games. At the start of the season, I think a lot of fans would be happy to see the second string and various youth players feature exclusively in the EFL Cup. It's a chance for others to step up, and for the first team to take an often needed break.
On top of fixture congestion, there's the financial and competitive incentive. Win the CL (or get to the final, even) and you get huge financial prizes (£15m+) and global recognition. Winning the PL speaks for itself. Get top four and you win big money (£30m+) as well as a chance to get back to the CL stage. In start contrast, win the EFL Cup and you get £100,000 prize money (that's from Wikipedia, surely can't be right?!) and a spot in the Europa League next season. Don't get me wrong, a trophy is a trophy, but for top sides those prizes are bordering on detrimental to the next campaign.
Similar problems surround the FA Cup. What used to be a great tournament has been left behind the times by poor decisions from the top. Whereas the PL and CL have dramatically transformed their profile and prestige over the last few decades, the FA Cup continues to go backwards. People (fans, players, managers) have been saying for years that the FA Cup should have no replays and that the semi finals should not be played at Wembley. There are opportunities to really improve the appeal of the competition, but bad decision after bad decision has kept it stagnant. Just imagine if the winner of the FA Cup grabbed a Champions League qualification spot? It would be transformed into must-win overnight. One of the biggest annoyances for me with the FA Cup is the media's incessant talk over how great it used to be and how we all grew up with it. It was great, I get it, and I've been to some fantastic cup games myself, but it isn't what it used to be, and quite frankly, as fun as cup upsets were back in the day, does anybody want to see Scunthorpe and Bristol in the final in this day and age?
Unless the format of the cups changes in the coming years, I can't see myself being all too bothered about them. Would it be nice to win the FA Cup this season? Yeah, sure, but it is definitely third on the list of priorities, behind a good finish in the Premier League and a strong run in the Europa League.