Destiny has nothing that could destroy precursor constructs, as the only thing that can destroy them is a halo.
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The dreadnoughts super weapon could destroy some of the precursors constructs because of it turning Oryx's throne world inside out. But I would imagine that the dreadnought won't last long against precursor star roads.
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The dreadnaught cannot destroy the precursor star roads because they run on neural physics. Forerunner fortress class ships (which make the dreadnaught look hilariously small) couldn't scratch them and they were only destroyed when the forerunners activated the halo rings
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Unless the dreadnaught works the same way as a Halo then I highly doubt it. I haven't brushed up on my halo lore in awhile, but last I knew they were completely invulnerable to all forms of damage outside of a Halo due to the precursors' mastery of neural physics I think is what it was.
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The dreadnought's super weapon is described as being like an oversoul's death impulse which could be considered similar to halo's effect , so it could be possible for it to destroy precursor tech but at a far shorter range.