I was attempting to disable the nametags of other players in spectator mode (by putting all spectators on the same team) using hideForOwnTeam to make the actual players in the game stand out more, and so the other spectators flying around aren't so distracting.

Toggling any option of nametagVisilbility does not work on someone in spectator mode. Name Tags are permanently stuck on for them.

You need 2 players in the world:

/gamemode 3
/scoreboard teams add test
/scoreboard teams join test @p
/scoreboard teams option nametagVisibility never
or /scoreboard teams option nametagVisibility hideForOtherTeams

or

/gamemode 3
/scoreboard teams add test
/scoreboard teams join test @a
/scoreboard teams option nametagVisibility hideForOwnTeam