It depends on the club and depends on the Cup.
back in the 90’s and early 2000’s the tops clubs did everything in their powers to fall out of the league cup ASAP. They usually did this by fielding an extremely weak team of youth players, reserves or bench players.
Wenger’s Arsenal made this obvious because they usually bowed out of the league cup in the first round.
If you are a top club its just not sustainable to play in 3 or 4 comps with a 25 man roster.
with that said, lower teams will make a run at the League Cup because it is a chance of silver wear and a place in Europe.
But all teams usually make an effort to advance in the FA CUP.
The national ice hockey league in North America (NHL) has been doing this for years. Former players are in the VAR studio assisting the referees in reviewing video.
In an interview, they said that former players are better at spotting cheating, such as dives or fouls with intent to injure.