There are no differences between:
Brainstorming first, finding nothing, FoWilling pitching Gifts and being stalled for a while
or
FoWilling pitching Gifts, drawing a card, Brainstorming into nothing and being stalled for a while.
The unlucky cards' configuration stacked on the deck, ruled the game for you, leaving almost nothing do think.
An intriguing ask could have been:
Would you have countered that drawer?While Brainstorming and then cracking the fetch to shuffle the shit, would be a non issue, I think that being sure of Countering that spell would have not been a simple choice.
The mana available on table for you is low, exactly as for your opponent.
If he is tapping out during your EoT, he could have a Bomb#2 hidden in his hand. Countering that spell would have saved him from probably losing resources or spells during his own Mainphase
If he is tapping out during his Mainphase, I cannot see any other good options in his hand, if not AT LEAST another land's drop and some other artifacts' mana accelerations, that would leave him with the opportunity of dealing with your own spells.
If he is tapping out during your Upkeep, he should have weighted that both you aren't going to be lethal that turn and you should have less counterspells without another card in hand.
If he is tapping out during your Mainphase, between the switch from 1st to 2nd mainphase, I can see a better plan. You seemed to have missed the land's drop, he realized that you are going to burn or not optimize the eventual ManaDrained mana, so he could evaluate to "risk" a Mana Drain when you are not well equipped to maximize his effect.
In a lot of situations, like the ones that I described, I would have not played my Drawer, fearing a nasty Mana Drain.
In a lot of situations, like the ones that I described, I would not have countered that Drawer.
Three mana on table and 3 or 4 cards in hand, with both hands, nearly untouched by Duresses or Counterwars, aren't so critical to manage from both the side of the table.
Both players should not consider themselves for a while, trying to better scuplting their own hands.
Bluffing an hand full of counters, when you are almost relying on a single Brainstorm, is really risky, especiallyif he is low on counters, such as you but with an hand with a couple of Bombs.
The drawer's typology itself,could be elaborated a bit more:
Brainstorms and Ancestral, both played at sorcery speed, would have been deadly and could only signify a Drain + FoW backup. It is clearly the worst situation available for you.
Clearly, TFKs or AKs or other overcosted drawers, are played at sorcery speed only if the opponent is dumb or protected by a couple of Pitch counters ( both FoWs and Mis-Ds ). They can be played quickly, without thinking too much about what the opponent can have, only after a deadly counter war or in the really first steps of the matches, but NOT when the opponent can have a Drain online.
I would have probably countered the TFKs only if it would have been played at sorcery speed or during my own 1st mainphase and only with a Mana Drain ( maybe drawn from Brainstorming into it, in order to optimize with the drained mana, the Gifts into my hand).
If none missed a single lands' drop and from the game situation described, the opponent went first because it have been the first one that tried to abuse of his slow hand, especially if the ask involve some EoT actions.
Going first, another time, revealed to be one of the "powerful stategies" available for T1.

Maxx