Oxford Advanced Learner's Dictionary

I can't find a duplicate or a †relevant definition in a respected dictionary (Cambridge Dictionary, Merriam-Webster, Collins, The American Heritage Dictionary ..., Dictionary.com, Wiktionary).

However, 'would' (among many other usages) is used to express ruefulness, resignation (after an unfortunate coincidence, say, with fate apparently mocking) in sentences like:

• It would have to be today she decides to visit, just when I've planned to spend a day painting the fence.
• And it would have to happen just when foreign policy has become so vital that Germany has a foreign minister that no one takes seriously .... [Jörg Lau - Die Zeit (Hamburg)]
• He would pick the only chocolate in the box I really like.
• The van was so old, the breakdown was inevitable. Of course, it would just happen to be in the middle of a busy roundabout.
• I would just choose this one day to leave it open. ['All Creatures, Great and Small'; Herriot]

Note that 'would' is stressed in all of these sentences, unless there is a better candidate (I have added the stress myself to the quote examples).

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†Perhaps Britannica Dictionary's sense 14 (also listed by Oxford Advanced Learner's Dictionary as their sense 13] is as close as I can find:

would [14]

used to express your displeasure with behavior that you think is typical of someone

• “He said that you were the one who made the mistake.” ... “He would say that, wouldn't he? What a jerk.”

OALD has a similar definition.

Longman possibly gets even closer, perhaps bracketing the usage in the question ('action' doesn't mandate a sentient agent, but is most usually used with one when not further specified, as in 'the action of rain on rocks' ... and the examples given are restricted to ones with sentient agents) with the displeasing but expected behaviour one:

*would [11]: typical behaviour spoken used to say that an action is typical or expected ... usually used to show disapproval

• You would go and spoil it, wouldn’t you! [emphasis mine]
• She insists that she did nothing wrong, but then she would say that, wouldn’t she? [emphasis mine]*

But fate is hardly equivalent to expected nasty behaviour.

Though perhaps personification crept in somewhere along the etymological chain.

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My AI Overview courtesy of Google for the more specific string "it would have to happen just when" is, as usual, spot on:

The phrase "it would have to happen just when" is an idiomatic expression used to comment on the inconvenient timing of an event, often with a tone of exasperation or irony. It highlights a coincidence where an undesirable or unexpected situation occurs precisely at a particularly inopportune moment. The expression is typically completed with a description of the specific circumstances that make the timing bad.

But, as illustrated, this sense of 'would' is far more versatile.