There shouldn't be a 'problem.' You'd think what you've described should really be the prevailing attitude amongst the majority of the clubs and their fans.
However, nowadays, most clubs, at all levels, seem to regard the early rounds of the League Cup, and EFL Trophy, as an unwanted inconvenience - except as a chance to blood youngsters and squad players - and only start treating them serious with strong team selections if their changed starting XI's manage to reach the later rounds. And those that take them seriously from the start are the exceptions rather than the norm.
I understand (but don't agree with) why managers do it. They know they're only really judged on league placings and so, so long as they're doing well in the league, then getting knocked out of cups early won't matter. But if they're struggling in the league, then a cup run will rarely save their jobs. Therefore, they prioritise the league games either side of the cup matches and rotate for the cup as that's the bonus 'nice if we win, not a big blow if we lose' game.
For as long as that attitude is also the prevailing one amongst fans, owners, and media, then, unfortunately, most managers will continue to rotate for the cups and focus on as high a league placing as possible for job security - rather than prioritise going for a rare piece of silverware. And the growing calls to abandon the 2nd (and 3rd) cups - already present before covid but gathering further pace now - will, sadly, continue.