I've a lot of sympathy for this type of thinking. I also think the lack of trophies, and easy win other teams supporters have in taunting about it, mean a lot of fans would put a League Cup win over qualification for the Europa League via the league.
Reality is we don't know what expectation the club have put on the manager. Maybe due to the financial implications, a higher league position is more valued. Maybe on a sporting level European qualification via coming 6th or 7th is seen as more do-able than winning the League Cup (where ultimately 1 or 2 Top 4 teams will need to be beaten in all likelihood).
Last night's result is always going to disappoint. How can it not? But other managers would be getting cut more slack after less than 10 games in charge. Not saying this is right or wrong but it's definitely true. It was always inevitable though and I think ultimately will be to the detriment of performances on the pitch. Any poor result is magnified by who the coach is. That will ultimately manifest itself in the crowd and in turn potentially affect the players.