In the first Thor movie it's established that the only people who can wield Mjolnir are those who are worthy of it. This is not tied in any way to ruling Asgard, that appears to be a lineage thing.
This line was broken when Loki died and Thor decided he didn't want it and therefore gave it to Valkyrie.
If Thor and Loki had both died and Steve had gone to New Asgard with Mjolnir, they would have most likely said thanks and then just chosen someone Asgardian to rule (probably still Valkyrie).