Your general premise isn't wrong, but Liverpool is owned by an American consortium and is competing just fine. It's the difference between doing it easy/fun (i.e., money and outlay mean absolutely nothing) and just being well-run.
In general, from what I've seen, Chelsea supporters have kind of adopted this bunker mentality where they see themselves as victims and resorted to a lot of whataboutism (some of it justified, some of it outrageous). They were on easy street for 20 years, now they may have to live in the top 15/20%, not the top 1%. Boo hoo.