Winning it is great, I'm always happy to win it. Going out in the quarters or semis however isn’t. With the CL, a run to the quarters or semis can be seen as a half-decent achievement and you probably knocked out a decent team or two getting there. With the FA Cup, it replaces the league for the first couple of weekends anyway so there's a certain novelty to it. The first few rounds of the FA Cup are still fun, even if the FA Cup isn't as important as it was.
But with the league cup, there's little interest in the early games, since its uncommon to play an interesting team before the semis. The only novelty value is giving your bench a run out. There's three extra mid-week games before Christmas, which never seem to come at a good time. For example, this year's 4th round fixture falls slap bang in the middle of our most difficult run of the year, where we play Leicester, Atalanta, Liverpool, Spurs, Atalanta then City in three weeks. The quarter finals are Christmas week this year for some reason, which is particularly shite. Then being in the final always shunts a game right to the end of the season, which means fixture congestion on the run in.
All of which is no problem at all if you actually win the thing. The final is a decent day out still & its a trophy. But if you lose in the final or the semis then all you've done is make your league campaign that much harder, and no-one will remember that run to the league cup semis of 2022.