First off, this is a very common confusion point for people learning theory. So you're not alone and it's not necessarily obvious!
The mistake you're making is conflating F# maj and F# Locrian. Remember that F# Locrian is the seventh mode of the G major scale, not a mode of F# Major. It is a different set of notes than F# major. Just like you listed the fourth mode of G major as C Lydian because it's the fourth note in the G major scale, F# is the seventh note in the G major scale, therefore, it is the seventh mode of G. So you will start on that seventh note (F#) to begin the pattern. Use that Locrian pattern from that F# note, then begin the Aeolian pattern, then the Mixolydian pattern. This will complete that Locrian sound for you.
The starting point is more or less your root note. I do have a semantics preference different to Tyler that might at least clarify. He says that playing a mode of G is still the "key of G". While this is technically correct because it has the same notes, it isn't necessarily how you talk in the music world. If you're playing E minor for example. We say we're in the key of E minor. We don't say we're playing E minor in the key of G. E minor (Aeolian) would be the sixth mode of the G major scale. But if we always say "key of G" then it becomes confusing and it's hard to differentiate. We don't always use the modal name for describing keys, but we generally at least use the root and minor/major. Using C Lydian in the example above, I would say I'm playing in C major. A lead player would then listen to the chord progression to determine which of those major modes (Ionian, Lydian, or Mixolydian) they could play to sound good on top of the chords! This part will be something you start seeing as you continue through Tyler's course. But understanding his semantics are important to really benefit from the course. So remember that Key of G = all modes encompassed within that scale. Even if the root is a C like in Lydian, it used the same notes. This part is hard to wrap our minds around because it's the same thing, but different.. The reason it sounds different is that we're emphasizing different tonal checkpoints. We traverse the fretboard differently. We have different chords behind those same notes. There are many other factors to how it "sounds" versus what is actually played. It's fascinating.
Again, hopefully I helped some. If I wasn't clear about something, don't hesitate to ask a follow up!