Your 1GHz op amp example isn't going to give you an ampere
of output. So forget it.
Your 1GHz (unity gain) op amp bandwidth, if you wanted any
decent gain error (1% = 40dB = A>100) pulls back to 10MHz
by the standard GBW product relation. So you'd be right
where you complain about, more or less, using that op amp
even neglecting its output-drive inadequacies.
Your op amp doesn't have anyone expecting that its load and
loop characteristics will change all over the place. Regulators
have to be sandbagged for the worst of it. That means things
like using explicit load capacitance and plenty of it, to be able
to ride out the interval between application of a step load or
input voltage, and the amplifier's phase lag.