Didn't Hades' sword yeet his soul to nothingness?
One confusion with this is that the original manga never describes any special qualities of Hades' sword.
Hades was going to eliminate Seiya and prevent him from reincarnating, but this seems to be an ability or power of the god and not a power of his sword.
It is never described that the curse will prevent Seiya from reincarnating, in ND it is only described that Seiya will die in three days and nothing else. We still don't know if his soul will be eliminated after those three days or if he will fall through the hole in the Yomotsu into the cycle of reincarnation, since Kurumada only says that Seiya will die in three days when the invisible sword (created by Hades' hatred) reaches his heart.
Comparing LW Seiya to ND Seiya is something people do a lot but the events in the LW are no where close to the same as ND
in ND Seiya should have the 8th sense which already removes you from the entire concept of the cycle of Samsara.
where as in the LW this Seiya probably only had the 7th sense which is highly unlikely considering Aiolos is alive int his time so the rebellion probably never happened. Making this Seiya probably only a 6th sense user.
So what Hades sword is doing to ND Seiya is akin to "Forcing the concept of death" back into Seiya; Forcing Seiya back into the Concept of Samsara so that he can be killed and removed from existence
Remember that Bronze Saints do not master the senses, therefore Seiya does not possess the 8th sense or even the 7th sense mastered in ND (although now, thanks to Hades' curse, even all his cosmos has been removed from this world, so he has no sense or trace of his cosmos), even Dohko says that Bronze Saints will only awaken the 8th sense for a few seconds.
That is why the Bronze Cloths returned to their normal form and again had trouble fighting the Gold Saints.
Also the curse of Hades eliminates all traces of power, cosmos and abilities of the opponent, in short it completely nullifies him, at this point Seiya is even less than an ordinary human (even Shun wonders if he is dead or alive when he sees him).
I mean, the other 3 bronzes had god cloths, so they must have been somewhat elite Saints
The Bronze Saints in that universe were so weak that they got into trouble and were killed with a god in a human body. The God Cloths in the original manga were almost indestructible, even the power of Thanatos (much more powerful than Poseidon in a human body) could not damage these armors and could withstand several attacks from a god like Hades at maximum (probably the most powerful enemy in the manga continuity). Also a detail to consider is that the real God Cloth in the original manga are golden, similar to the armors of Athena and Artemis, that's why Thanatos confuses these armors with a Kamei.