The geography of the world in ds3 is completely incompatable woth the world of ds1. And not just a little bit. Areas that are next to one another in ds1 are on completely opposite ends of the map, with entire mountain ranges and kingdoms in between. kingdoms that don't even exist in the same time period. furthermore, the break down of time and space isn't a sudden event. its a gradual phenomenon. hence why you see the sun bleeding red at the end. by the time ds3 starts, the world is already on the brink of total collapse. your explanation makes no sense.Different kingdoms from different eras are only smashed together around the kiln of the first flame, and the Ringed City DLC, which is not Lothric proper. Its very explicitly distinct, as made clear by the Firelink Shrine becoming an alternate version of itself.
The most common stated explanation is that the Ornstein you fight in DS1 is an illusion conjured by Gwyndolin, just like the Sentinels. This is supported by DS3 stating that Smough alone was the last knight to stand in defence of Anor Londo.
You find Ornstein's armour in Archdragon Peak because he travelled there to seek Gwyn's son. Its possible that the "Dragonslayer" you battle in DS2 is someone using his armour, just as its stated that Havel's followers also use his armour. Its also possible that Ornstein is undead, and returns from death after DS2 (just like you do) to resume his travels.
I mean, its vague as all hell, but plausible explanations exist.
as for ornstein, it also makes no sense. when you kill an illusion, you don't get a unique soul. not once, in all of ds1. think of gwynevere. during the O&S boss, the one who dies gets absorbed by the other. you then get the others soul. you are able to get ornsteins soul. and unlike other characters in ds1, whose deaths were optional, you have to kill ornstein. ds3 flat out says that ornstein left. it's either a ret con, or ds1 and ds3 take place in alternative time lines. both are disgraceful examples of bad writing. ds1 wasn't perfect, but the writing got consistantly worse with every dark souls game.