In some systems, players can create dramatic contrivances too. That only requires giving them a metagame tool for it (or just a GM open to player's authorial input).
In my 4e games I tend to handle this via skill checks (eg Streetwise for fortuitous meetings). Not an especially elegant solution, but it doesn't come up that often - I mostly just use GM fiat in response to players' expressed interests.
Eg One of the PCs has, from his first appearance in the campaign at 3rd level, been part of The Order of the Bat, a mostly drow secret society of Corellon worshippers dedicated to undoing the sundering of the elves. (In case you're wondering, yes, the PC is a drow and the player made up this particular bit of backstory!) The first time the PCs met some elvish NPCs - surface elves, not drow - the player of the drow PC declared that he was making the secret sign of The Order of the Bat, hoping to achieve recognition from the captain of the elves. I can't remember if I called for any sort of role or merely fiated it - I think the latter - but in any event the upshot was that the captain paid no attention to the signal, but the lieutenant noticed it, and gave the ritual response (if my memory of fiating it is correct, my reasoning will have been that a secret alliance with the captain might be a bit too good, but with the lieutenant was useful without being too good). The player was then able to have the lieutenant take a dragon's tooth so that it could be forged into a Wyrmtooth Blade - not a power up (just part of the usual treasure parcel system) but creating an in-fiction rationale for how such an item could be acquired.
In MHRP, meeting a helpful contact simply requires paying a plot (= fate) point, and grants a bonus die to the pool if relevant. Meeting enemies is handled by the GM!
In Burning Wheel, fortuitous encounters are handled via the Circles mechanic. In the example I gave in my OP, after seeing the arrival of the holy man Bernard's ship the player of the rot-afflicted PC made a Circles check to meet a cleric in his entourage and succeeded, and hence encoutnered his childhood friend Jessica (who had gone off to religious training when the PC went off to mage training) among the holy man's entourage. (If the Circles check is failed, the GM is entitled to introduce a hostile NPC instead - and it was a failed Circles check in the campaign's first session that established the PC's nemesis. This sort of thing is what makes BW a harder, grittier game than something like 4e.)