Empowering a prophecy without railroading, using mechanics, is very much possible. The "stick" is not so useful here as is luring the player with the "carrot". There are a number of more-or-less successful games that do this with great success*, to the point that some use it as a central part of character development, so there's evidence in the wild that this is possible.
All the games I've observed doing this effectively offer a significant "bribe" that tempts players to choose to act in line with their fate. Such a bribe must be large enough to give the player a true dilemma – do I act with freedom to choose the "best" course of action for my character, or do I take the big mechanical boost and do what the prophecy says I must?
The bribe has to be big, unbalancing almost, in order to make it large enough to be the equal to how much a player values their freedom of choice. The extra that they're getting must be valuable enough to really make them consider it. In D&D terms, a +5 or more to all skill and hit rolls for the duration of the time where they are facing the direct consequences of their choice (the resulting battle, the escape from the crypt, etc.) might be suitable. Another effective carrot is the game's improvement currency: a large XP bonus (perhaps 10%, or more, of what they need to level) can be very tempting for a player.
This can be tweaked as you go, to. You don't have to tell them at the outset that every such moment is worth 1000 XP — instead, when they come to a pivotal moment in their fate, you can mark it as such by saying "…and if you choose X against your better judgement, it's worth N experience." You can offer variable carrots tailored to the importance of the choice, too, so that you give bigger bribes to properly balance the sides of the dilemma.
It must always be a free choice. You might set the dilemma, but a player hugely values their freedom to choose.
By putting the choice into your player's hands, you make it more likely that they will fulfill parts of the prophecy/curse/fate and you don't railroad it. As long as you're always prepared to accept the player's true choice (don't up the bribe after they refuse, for example), they will feel it really is their free choice, and you'll get much more buy-in for when they do choose to accept the prophecy.
Not everyone likes talking mechanics
The one caveat is that some players who go for deep immersion will really dislike baldly discussing mechanics when they're trying to choose according to their character. There's no real good way around this without seriously annoying the player and undermining their buy-in. The compromise application of this idea is to not tell them that they have the choice – but when they do choose the path of prophecy, narrate how their sword swings truer (as you start appling a hit bonus to their rolls) or how a great weight of foreboding settles in their chest (as you give them bonus XP as an aside). This will create the association between bonuses and accepting their fate, without interrupting their internal narration and breaking their suspension of disbelief.
* One example is The Riddle of Steel. When you're acting in line with your Fate during a pivotal moment in the prophecy, you get a very large bonus to all your rolls during the fallout of the choice. Another is Dungeon World, where you can get into situations where if you do a thing that you normally wouldn't choose, you get to mark XP.