Don't think grieving, think religious ritual. A lot of the more formal or ritualised activities that you see described in Homer are just that, rituals.

And I don't just mean the obvious things like sacrifice scenes. Hospitality scenes, too, are essentially a transferral of a religious rite called theoxenia into the mortal realm: in theoxenia, people ritually welcome the god in, set a place for the god at a sacrificial feast, and follow it up with liturgical prayers and gift-exchange; in Homeric hospitality scenes, people formally welcome the guest in, set a place for the guest at a feast, and follow it up with formalised questions and gift-exchange. It helps that some of the most important of these scenes in heroic epic actually have a real god as the guest (in disguise) -- Athena in Odyssey books 1 and 3, and Demeter in the Hymn to Demeter. It may well be the same with mock battles at religious festivals, like the Eleusinian balletys: we don't know much about the balletys beyond the name, but I'd be very unsurprised to find out that it mirrored Homeric battle scenes in some ways -- or rather, that poetic battle scenes emulate the religious rite.

It's the same with athletic games. In real life these were an integral part of religious festivals: not just the famous ones at Delphi and Olympia, but also games held at shrines in honour of heroes (semi-divinised legendary figures of the past). Now, the key thing about a hero shrine is that there's a tomb for the hero there. So stories about the hero's death, or which emphasise the hero's mortality, are prone to being linked to hero-cults; and since you hold athletic games as part of a religious festival in honour of dead heroes, it makes sense to transfer that back into the heroic narratives that the cult is based on. The result is that you have legends featuring athletic games at a hero's funeral.

Now, I wouldn't rule out a certain amount of re-transference of the legendary athletic games back into real-life practice. If so, you would end up with: (a) real-life funerals with athletic games to emphasise the greatness of a deceased real-life person, modelled on (b) legendary athletic games held at legendary funerals, which are in turn modelled on (c) real-life religious festivals with real-life athletic games in honour of legendary heroes.

I don't know my way around funerary ritual well enough to cite any late examples of this kind of meta-meta-ritual, later than, say, the sixth century. But one good contender might be the funeral of Amphidamas, who was supposedly a hero in the Lelantine War (which may or may not have been a real war), at which there were held funeral games and a rap-battle between Homer and Hesiod (who may or may not have been real people). It'd be a bit much to take all of that at face value, but, you know, bits of it are at least possible.