Why do we need fridges if we can just put things in the landfill and have it last longer

Hard to believe was what was being asked

Not really, Hard to believe but true was the question - but there's no evidence or even theory that this one may be true.

Ordinarily I'm not a super nitpicky asshole, but given the 100% true requirement it feels kind of appropriate?

the number is present in many websites about trash or composting

The number of websites quoting the same thing makes it seem credible but ultimately doesn't mean much. All I can find is blog-spam quoting this number. Suppose I write on my blog that lettuce decays in 25 days.

Even if this is exaggerating, the moral of the story is [...]

Obviously, we shouldn't be putting vegetable matter in landfill, but hyperbole has ever been a very poor way to promote ones cause.

But it’s possible that that lettuce was a fresh and plastic wrapped thrown to the landfill like that, because that does happen as well. And maybe that created optimal conditions to prevent decomposition.

The anaerobes are already present on the lettuce from day 1. Wrapping in plastic doesn't keep them out because they're already in. Even if you could create optimal conditions for preserving lettuce in landfill, that's not really the same as the claim which is being made here.

I really don't know anything about this but ... It looks as though most vegetables will be 100% sludge after 8 weeks in anaerobic conditions.

I suspect that this 25 year factoid is derived from some specific conditions and very specific use of the term "completely decomposed". For example, maybe under laboratory conditions, anaerobic decomposition quickly turns a lettuce to sludge in 8 weeks, but can't break down cellulose polymer chains or something. So for all intents and purposes the lettuce is gone but if you looked at the sludge 20 years later you would still find some of those polymer chains present.