I was looking back at Kid Icarus: Uprising because I was bored and I had the sudden realization that Smash Run was likely only possible at all because most of its enemies (and really just most of its assets in general) are just ripped directly from that game.
And, similarly, the reason it was not in the Wii U version was because the models would be too low-res to justify putting it on a home console.
Continuing on with this line of reasoning, the reason it is not in Ultimate is because they'd need to design and create a bunch of new, high-definition assets, which likely would have taken up too much of the development time for Spirits mode to be implemented.
As such, there will sadly likely not be a Smash Run where we can use Spirits coming in an update to Ultimate, ever.