The ancient problem was how to model modality, Parmenides' world was necessary through and through, no contingency: if only what existed existed then it necessarily existed, was the intuition. All there is is the set of actual things. Possible worlds modelling, Leibniz, Frege, showed how to formalize necessity, contingency, possibility, impossibility by designating all actual things in all possible worlds. Modal realism of Lewis is thus nominalist strategy (slightly ironic since nominalism aspires to ontological austerity). Alternatively Plantinga claims that we can be actualists (only the actual world exists) and can still have a metaphysics of modality by positing that (or is it having a revelation that?) immaterial, mind- and matter-independent Platonic entities, such as essences, properties, propositions and states of affairs, are taken to exist in addition to matter. The nominalists about universals are typically nominalists as well about the philosophy of time: Particulars have temporal parts (are spread across time) just as they have spatial parts. Thus you never really change: we just experience different time-slices of you. Like all particulars, on this view you are a spacetime worm. That's how the omniscient god sees you, looking down on all of time spread before him like a plane: you're a spaghetti-like thing stretched across it. Notice this parallels the way you are smeared out across possible worlds on the nominalist view. On this view, all points in time are equally real (like all points in space).
The Platonists hold that objects in time are wholly present at each moment in time. They identify the particular with a form ("substance") and thus have no problem about the identity of the particular changing because constituent matter is changing. For the same reason they have no problem with holding that only the present moment exists (Aristotle in a nominalist mood argued that only present moments did not exist, as "moments" are conventional boundaries of divisible amounts of time, and so past times are bounded by past moments, future times are bounded by future moments, and "present" time periods are bounded by one past moment and one future moment.)
The problem of fate is really just a version of the problem of modality. It's really more a problem about the present than the future. Notice that both the Platonist and the nominalist claim to offer solutions to the problem: they both claim they can explain what we mean when we talk about necessity, contingency, and probability. But both schools have to adapt and revise to do it. Nominalists initially insist that only concrete particular things exist, and later develop modal realism to try to avoid the fatalistic implications of their original position. Platonists initially insist that eternal and unchanging Platonic entities shape the material world, and later try to adapt this ontology to account for modality.
So one question is, which feels freer, the nominalist view of time or the Platonist view of time? (Nominalist view is "B-series" re McTaggert, Platonist view "A-series"; that is, B-series refers to the model of time as a dimension, with all points equally real, while A-series sees time as moving through the present). Nominalist says tenses are indexicals. Platonist says tenses are metaphysically significant.