I always interpreted that scene as Somnus just testing Nyx's selflessness. He probably would have burned if he chose to selfishly sacrifice someone else but because he opted to sacrifice himself he was granted the full power of the ring. From the Kings of Yore's perspective it would be a bad idea to give that power to someone who is only looking out for themselves and trying to weasel their way out of accepting the full burden of the responsibility that comes with the ring's power. We see this concept come into usage again with Ignis when the ring constantly asks him if he only wants to risk death or outright sacrifice everything for Noct. Up until he lets go of that last shred of hesitance to straight up die the ring doesn't grant him its full power. The only difference there is that Ignis doesn't immediately burn for not going all in which can probably be attributed to the fact that he is already fiercely loyal to the chosen king which prompts the Lucii to give him some more leeway.
He was testing Nyx's selfishness. Somnus acted selfishly, and then when acquiring the ring, learned of his role. Ultimately, it seems all 13 Kings of Yore were significant in that they had great guilt due to a massive tragedy in their tenure. Somnus, specifically, was the betrayal of his own brother for his own gain. Nyx underwent a test of selfishness, but ultimately it's because Somnus sorely wants to see someone do better than he did. Nyx didn't sacrifice Libertus or Luna, or try to even highlight their safety. Rather than letting his own personal feelings get involved, Nyx sought out a future that didn't see him betraying two people close to him. Nyx didn't even consider it, whereas Somnus was consumed with guilt for what he did to Ardyn, someone who like Nyx didn't care about the semantics, merely about the greater good.
If Somnus denied Nyx the power of the ring after his little speech, Somnus would be denying someone less selfish than him the power he himself stole. It's both a test of Nyx's character and selfishness, but ultimately, moreso Somnus wants someone more brave than him. Nyx merely sought the diversion Regis wanted, so the big Insomnia battle happened and Nyx, merely a refugee, died. Ignis himself was dedicated to the Chosen King and given he didn't burn through the magic of the ring like Nyx, and due to his proximity, they likely did allow Ignis to live to protect Noctis. However, given Ignis's main desire (as told by the theme of his alternate ending, "My Humblest Desire"), was to
see Noct grow up as king. The price for using the ring was his greatest desire. Nyx wanted a future for people like his sister to see, so he was robbed of the chance to ultimately live for it. Somnus seems to be of the belief that those who use the crystal and ring are typically those that must atone for using such power (likely a Bahamut thing). As such, there is always a price to pay for using the crystal, the kings using their own lifespan. However, the Lucii and those worthy of the ring tend to have some level of selfish desire that may be exploited, and ultimately while Ignis did want to protect Noctis, it was because of his relationship more than his care of the future (he even states Ardyn could do whatever with the world in the alternate ending). As such, Ignis paid with his desire for Noct to live his life.
Bahamut seems to be of the opinion that due to mankind building the Omega Weapon, they must die and as such, even their protectors in Lucis must also bleed and suffer with their own lives and even undeath as the Lucii remain unable to really pass on due to their accomplishments, typically involving a shitload of people dying and suffering.
