Steam Knight would not trigger if the aura was activated by Final Gate.
The parts of a turn are described in https://2e.aonprd.com/Rules.aspx?ID=435
"Step 1: Start of Turn" allows you to use 1 free action during the start of turn, but the free action must have a trigger of "Your turn begins” or something similar, which Final Gate does not.
Consequently, Final Gate must occur during "Step 2: Act" with all other actions whose trigger is not explicitly "start of turn". This is after all start of turn effects which are checked in "Step 1: Start of Turn" have already resolved (or been skipped if their trigger condition was not met), including Steam Knight.
Since Steam Knight checks whether your aura is active in Step 1, but Final Gate does not turn your aura on until the very beginning of Step 2, Steam Knight cannot resolve after Final Gate within the same turn.
Note: the argument could be made that "the first action of your turn" necessarily implies Final Gate must happen in Step 1, since any action taken in Step 2 wouldn't be the first action of your turn if any other action were taken in Step 1. I don't think that reading is the most natural reading of the rules, but I understand it.
In this case Steam Knight still cannot trigger in Step 1 if the aura is activated by Final Gate in step 1, since you can only take one "Step 1: Start of Turn" free action.
If Final Gate occurring during Step 1 or 2 is important for any other interactions, it's a good idea for the DM and player to discuss this interaction ahead of time if a player is planning to play a Kineticist so everyone is on the same page.