Learn English – would always have OR would have always

Yes, this is acceptable.

Both your sentences present conditionals; although they are not expressed as formal IF ... THEN constructions, they might easily be so:

The situation would have been worse if he hadn't helped.
It would have caused us a lot of problems if anybody had not cooperated.

The protasis (IF clause) of a conditional may lie outside the sentence at hand; this does not invalidate the use of would:

It looked like the school might have to close. That would have been a shame. = If the school had had to close, it would have been a shame.

All of these uses are equally valid with non-past^✲ reference:

The situation would be worse without his help.
The lack of anybody's cooperation would cause us a lot of problems.
It looks like the school may have to close. That would be a shame.

In fact, most uses of would may be understood as implicitly conditional, though the actual condition implied may be very vague:

I would like a piece of pie. [eg, if you have any or if you would be so kind as to bring me one]
I would be happy to do that for you. [eg, if you were to ask me or if you need help]
I would never say such a thing! [eg, if the opportunity arose or even if I were provoked]

^✲ I should perhaps note that past modal + have + past participle , although it looks like a “perfect” construction, has nothing to do with the ordinary English “perfect”. It is, rather, the language’s workaround for the awkward fact that the modals can, may, shall, will may be used in their “past” forms with non-past (present or future) reference—which leaves them no convenient form for expressing the same modality with past reference. To accomplish that, we employ the “perfect” construction. It would have been is in effect the simple past of It would be.

Some people might think OP's original text using the Perfect Infinitive (to have seen) is more "logical" because it more explicitly echoes the past tense element of would have liked. But note this from Garner's Modern American Usage (2009)...

would have liked
...should invariably be followed by a present-tense infinitive — hence would have liked to go, would have liked to read, not ✳would have liked to have gone, ✳would have liked to have read.

Having said that, Garner himself acknowledges that the erroneous phrasings are very common. In fact, Google Books claims 386,000 written instances of would have liked to have seen. That's not so common as the "correct" version would have liked to see with 1,050,000 instances, but I think it's enough to justify saying that unless you need to pass a "fussy" exam, you could reasonably call it a stylistic choice.

OP's "Simple Past" version ✳I would have liked to saw is idiomatically and grammatically unacceptable.

EDIT: I see the question has been edited to tell us that the supposedly "incorrect" (according to Garner) version actually comes from a BBC Learning English page. I take this as further evidence (if it were needed) that Garner's position is unjustifiably pedantic/prescriptive. And comparing American and British charts on Google NGrams, I see no evidence that his position reflects any kind of AmE/BrE usage split.

At the risk of stoking controversy on what I consider to be something of a non-issue, I'll just cite this from grammarphobia, who also seem to have little time for Garner's position...

Using two “haves” (as in, “I would have liked to have gone”) is usually incorrect, because it’s unlikely that you really intend to talk about two separate times in the past.