@HS You see, I keep my posts undetailed and uninteresting, just so that WW may shine

@WW agreed, "fossilized" was the wrong word.
You're absolutely spot on in saying that "when we took 'de' from French, we took on a lot more than possession and partitives", and this is the root of the question: which usages are not mirrored by the English inflected genitive.
So the short list of exceptions would now be: partitives, nouns of thinking or feeling, and the CaGEL fossils.

"a body of evidence",, "a can of beer", "a mass of documents" = partitives; hence we cannot use a genitive (not "a document's mass).

This leaves us with expressions like "a man of culture and sensitivity", "a part of speech", "the axis of rotation" ,, where we are faced with a faintly dated use of "of" to denote a quality or characteristic in a phrase which in modern English might equally be expressed either adjectivally or as a compound noun: "a cultured and sensitive man"; "a word class"; "the rotational axis", (but seldom, "the rotation's axis")

"a sense of pride", "a feeling of despair" seem hard to explain as other than fossilized expressions

errata: "the lions' slaughter" -> not clear whether the lions died or they killed a lot of anmals

A few more idiomatic items:
"the sweet smell of success" vs "success's sweet smell"
"the stench of failure" vs "failure's stench" (but: "failure's foul stench")
"the state of the nation" vs "the nation's state"
"a sense of pride", "a feeling of despair" .....

Re 4) objective/subjective genitives: this really only comes into play if the verb-from-the-noun is possibly transitive; thus:
"Tom's death" -> die is intransitive therefore Tom is the do-er and he is dead.
"his sister's murder" -> did she die or was she murdered? Should be clear from the context, unless of course she killed someone and was then herself killed.
"the lions' slaughter" -> not clear whether the lions died or someone slaughtered them.
"the slaughter of the lions" -> prima facie suggests it is the lions who died
"the shaft's rotation" -> no distinction with ergative verbs
Thus as the genitive simply denotes some relationship, we have to pick up the meaning from the context.

No hard and fast rule here, but as general guidelines I would suggest:
1) be wary of genitives to indicate composition, : "a book of leather" not "leather's book";

2) attributes seem to be more idiomatic: "a man of honour" not "honour's man", but "a woman's scent", "at death's door"; but again "he was awarded the title of President" not "the President's title"

3) use the adjective or compound noun where appropriate eg the presidential title, engine oil

4) be wary of objective genititves: "the love of music" not "music's love"; generally 'a woman's love" refers to a woman doing the loving, whereas 'the love of a woman' is more ambiguous.

@WW Thanks: I never realized that "Magna Carta" was a Southener thing; and I always thought of "Swing" as a music genre.
"He was sat" is hard to explain grammatically; just idiomatic.

One of the little hurdles with a descriptive approach to grammar is we can end up talking about usage without providing a good explanation of how it got to be like it is.

For example we know "may" can be about possiblity or permission/prohibition and one just has to pick up the relevant meaning from the context. The possibility meaning exists in the equivalent German verb (es mag sein); but I would love to know how we came to permission/prohibition.

Similarly, I am not entirely convinced that "could" in the following sentence MUST be replaced by "were able to" (cf Hewins ) , although it is doubtless less common or less clear:
"..... then it started snowing heaviliy; but despite the blizzard we could drive home safely". Again what is missing for me here is some explanation; I cannot believe it was so complicated in OE or even ME

1393, 1489, 1936, the North beyond Edgware was ever non-conformist

From a grammatical standpoint one can use any noun as a verb if the meaning is clear in context; whether it is good style is another matter.

Thus one can "pen" a letter and so on. The exception to this is where a separate verb form exists already: I can inform you, but not information you.

Leverage as a verb is widely used in financial circles when referring to gearing or the debt/equity ratio. Outside of financial circles the meaning is metaphorical and perhaps just a fashionable buzz word