not true.
there are plenty of constructions in which that's ok. example: he took annette's growling dog out for a walk.
no problem there.

the issue here is just that "labor's unqualifying support" is just ridiculously awkward. this is one of those things that native speakers will understand almost instinctively, but that is nevertheless nearly impossible to explain to non-natives.

in general, i'm loath to use apostrophe + "s" for anything but humans, animals, and the like. this is definitely NOT a hard and fast rule, but i've noticed that it's fairly consistent across most usage.
thus, "an ape's vocal tract" is preferred to "the vocal tract of an ape", but "the colors of the mural" is better than "the mural's colors".

in general, if you get to pick between the apostrophe+s construction and the "of" construction, and the possessor isn't a living thing, i'd go with the latter.
but by all means try to eliminate based on other things first.

the REAL issue, though, is "unqualifying" -- this is incorrect.
"unqualifying" means "not meeting some sort of standard for qualification".
the intended meaning here is "unqualified", which means "without any sort of restriction or reservation".

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the past perfect makes perfect (heh) sense here. the idea is that, lately, the women have been receiving support from labor.
the present participial form "are receiving" would also make sense, but remember that you aren't supposed to change the meaning of the sentence without a good reason for doing so.

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once, just once, i'd like to see a sentence about people dedicated to a conservative cause.
just once.

i think i'm going to have to wait a very long time.