Anyway, Lv. 9 comparison (Fighter 6/X 3). Both Variant Humans. Assuming Assassin gets advantage but no surprise. Assuming Gloom Stalker does not get advantage. All builds take Archery Style and Precision Attack and will attack AC 16 with Sharpshooter trade active (45% hit chance for DEX 16 build, 50% for DEX 18, 55% for DEX 20). Both use Precision Attack when they think they missed by 4 or less (so 20% of the time):
Case 1, longbows
Assassin 3 version took Alert, Sharpshooter, and +2 DEX (18) in order.
Gloom Stalker 3 version took Sharpshooter, +2 DEX and +2 DEX (20) in order.
Assassin:
(.75 + .2*.225)*2*18.5 + (1-(1-.5)^4)*7 = 36
Action Surge (the first part, minus Sneak Attack, gets doubled) = 65.8
Gloom Stalker:
(.55 + .2*.225)*(2*19.5 + 24) = 37.5
Action Surge (whole thing gets doubled) = 75
Case 2, Crossbow Expert at Lv. 1, hand crossbow
Assassin 3 took Alert and Sharpshooter after in order (stays 16 DEX).
Gloom Stalker 3 took +2 DEX (18) and Sharpshooter after in order.
Assassin:
(.6975 + .2*.225)*3*16.5 + (1-(1-.45)^6)*7 = 43.6
Action Surge (extra 2 crossbow attacks) = 68.1
Gloom Stalker:
(.5 + .2*.225)*(3*17.5 + 22) = 40.6
Action Surge (extra Gloom Stalker attack and 2 regular attacks) = 71.7
In any case, the two are pretty comparable at Lv. 9. Gloom Stalker will have a slightly better Action Surge all around.