Lot of good comments cleared up your question but keep this in mind.
The traveler battled the darkness, presumably while still in orbit since otherwise, it should have crashed on earth, unless it did in the edz. Either way the battle was close enough to have effects on earth if we believe the (non reef born )awoken where created on earth.
The distibutory the reef born awoken are from exists in a dimension where time moves faster then ours so a reef born awoken found by our ghost could have crashed attempting to land on earth or during any battle after the collapse. Remember the wall has stood for a long time and was built some time after the colapse and the rissen, whom came after the incident with mara and the singularity. It isnt known how long we where dead though admittedly quite a long time.
Lastly, while destiny's lore is often criticised for its often large room for interpretation, it does at least leave room for a wide possibilities to our origin.
Edit:
One more thing. It is never stated that what created the awoken could only effect human dna during life. You could have died human and been rezed as an awoken, and with no memory to confuse you, youd never know
English
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Awoken fallen exos.