It's a very tough question to be honest. I remember winning leagues based on getting a goal or two ahead and then killing the match stone dead. Happy to win and loved the parades at the end of the season.
As far as going out of the FAC is concerned, I don't think any LFC fan has said they were actually happy about it (we'd still like to have won it) but a pragmatic approach is to point out the undeniable (even by you) benefits to the club, as far as this season is concerned, of not being in it any longer. Even the blindest fan of any ilk should be able to acknowledge how important to Liverpool / Spurs / Arsenal staying in the Top 4 has become and that fewer matches not related to the PL enhances their chances..
However maybe it's because there have been few and far realistic chances of us winning trophies in recent years that playing entertaining football becomes a crutch of sorts, even though it's great to watch and maybe I too would take it over winning the FAC or LC or EL .. but not over the CL or PL when it's been so many years since we've lifted the league title.
Can't deny how enjoyable it is to look forward to every game though, even when not all pan out, as is the nature of the sport.
However conversely I think that maybe the memory of the glorious years under Ferguson have invaded and perverted the thoughts and opinions of United fans, who have been so used to glory that the thought of being without it and of someone actually just enjoying football for football's sake (which let's face it, the vast majority of football fans do week in week out, since the teams they support have no chance of ever winning anything of any real importance) is almost inconceivable and so find this attitude not only implausible, but even deny it's possibility.
And stop trying to suggest, with your final two sentences, that just because one, two or a few Liverpool fans have that viewpoint that they are representative of all Liverpool fans. Behave yourself.