#11 " It almost seems as though you're saying it's impossible to assert truth in a subordinate clause"
I definitely did not mean to give that impression!
What I said was
"presents a proposition about what is important and may imply the truth of the second clause (which could almost use a subjunctive)" [extra emphasis added]
In fact, I think, a that clause very often implies the truth of the content (or allows it to be inferred); some of the context may even make it the default meaning, but the line between meaning and implicature can be fuzzy.
Following a verb that expresses importance, exhortation, etc., however, I think it is less likely to be inferred.
I think a that clause (perhaps less so than a non-finite construction) merely states or designates a proposition, without necessarily asserting its truth. The degree to which an assertion is implied may depend on the verb form (present indicative, backshifted / modal preterite, "present" subjunctive, etc.)
"What matters is: climate change is man-made." [speaker asserts truth]
"That climate change is not man-made is a major element of the President's platform." [speaker does not assert truth].
"I do not believe that climate change is man-made." [speaker does not assert truth].
"What matters is that he does something about it." [speaker does not assert truth, but may be ambiguous]
? "What matters is that he do something about it." [speaker does not assert truth; I'm not sure if the subjunctive is idiomatic after a phrase like "what matters"]
"What matters is that he should do something about it." [changes meaning by adding more explicit deontic modality]