You personally can sell any Treasuries you have at par value because you are small enough to not test the liquidity of the UST market.
So BTFP is of zero use to you. You would be stupid to use it, even if you had access to it.
An institution is big, and selling at par might not be possible if the liquidity of the bid side of the book isn’t strong enough to support their selling.
The Fed is taking a precaution in the event that UST liquidity is too weak tomorrow to accommodate orderly liquidation of bonds to finance deposit withdrawals at some banks.