God Cloths won't have the ability
The regeneration and immortality negation if they have that, because the God Cloths are sacred treasures (it is an armor infinitely close to a Kamei and has the cosmos and the divine protection of Athena by the blood of the goddess, therefore it grants the power to kill and hurt the immortal gods) and the attacks of Seiya and company killed Hypnos and Thanatos forever. The only God Saint who does not have this power is Tenma, because even he could not destroy a minor god forever like Morpheus (the god's soul still existed after his attack) and neither did he completely eliminate his mother (in the Gaiden Aries it is mentioned that her soul and she still existed in the Tenkai/Olimpo).
Although Hypnos and Thanatos never showed resistance to regeneration and immortality negation.
In addition, Ikki should also get immortality and regeneration negation even without the God Cloth, because he now has the Sword of Chrysus as equipment.
Kairos had his power sealed iirc so you are right. He shouldn't get the ability.
The seal of Kairos did not seal his power, the effect was different and forced him to reincarnate as a human. The only limitation of his power is that he is in a human body and cannot change bodies like Thanatos, Hypnos, Hades and Poseidon. The seal puts him in a similar situation to Athena who also reincarnates as a human.
If he was eliminated so easily is because he is a weak god, that's why he considers Pegasus' power as impressive and a god killer (when Pegasus never defeated a great god in the manga, even Tenma couldn't completely defeat a minor god), and Tenma is probably one of the weakest Holy Gods in the franchise, because he was defeated by Alone and could do nothing against Hades when he controls his avatar (even Seiya with Gold Cloth fights better against Poseidon, who is much more powerful than the Hades of that holy war).
The plan of Kairos was to use the power of Pegasus to invade Olympus... three seconds later, Tenma was defeated and shattered by the first Olympian God in his path. The dumbest plan in Saint Seiya.
Iirc Abel had a way to kill Athena didn't he? Mars was going to kill athena by draining her Cosmo away iirc. Abzu was going to straight up absorb her. Aborsption is a big different, but Abzu did have an all out war with all the gods in the history of Omega. So he should get a "Possibly" rating. Didn't Odin create Draupnir?
These gods have Regeneration and Immortality negation. Abzu can eliminate anything in his universe because he is the origin of everything (the void or darkness that gave birth to the universe). And I don't remember if Odin is described as the creator of the Draupnir.
The problem is the resistance to Regeneration and Immortality negation.
Even if the gods fought it doesn't mean that they will try to eliminate each other forever.
Honestly idk why Tartarus has a profile. I tried to get it deleted but apparently he qualifies for a profile. Gaia is Gaea right? she was the one that created weapons that could kill the gods.
The weapons of the Titans (weapons created by Gaea) never showed the power to kill forever immortal gods such as Athena, Hades, Poseidon, Thanatos, etc. Although Gaea is one of the most powerful gods in the franchise, therefore, she must have the power to kill other gods forever.
I think Hades, Poseidon, and Zeus should qualify as well. They have had many holy wars against each other but they keep coming back. Typhon should also have it. I could missing some character here. not in a good spot to think at this moment lol.
Hades, Poseidon and Zeus (main universe version, because the version of other universe is unknown if it is as powerful) probably have that resistance, because they are gods of Athena's level, but they don't have the feat. And Athena is a good goddess, she almost never tries to eliminate completely the opponent, even she did not eliminate Asclepius and only destroyed his body and trapped his soul in the Tartarus, an opponent that probably will be defeated with the arrow of the goddess (a sacred treasure of Athena).
Athena only destroys Typhon's body, but does not destroy his soul and cannot regenerate his body, even his purpose was to get a new body, therefore, this means that he has no resistance to regeneration negation, because he cannot regenerate his body after being destroyed by Athena. This is similar to what happened to Asclepius in the last chapters of Next Dimension, Athena and the gods destroyed his body and threw his soul into Tartarus, so now he tries to get Odysseus' body to return to the world.
Although this is also an example of regeneration negation, because Asclepius cannot regenerate his body, although Odysseus with the power of the Golden Ophiuchus (the power of Asclepius) can regenerate his body from dust.