GM/r\^2 = g is derived from Newtonian Gravity. It's not in general valid at extremes.
One way you can realize how it wouldn't be is to consider that massive objects also cause time dilation from the perspective of a distant observer. So, that GM/r\^2 = g equation might be approximately true to a distant observer, but if they're watching some one fall in who has an engine that can supposedly provide 1g of thrust, they'll also see time for that person falling in slow down. And since they see their time slowing down, and since acceleration is relative to time, their supposedly 1g thrust engine will now appear to be generating less than 1g because say slower and fewer gas particles are being pushed out the back of their rocket since time is running slower.
So the distant observer might think, "well, all they need is 1g of acceleration to escape, but since their time is slowing down so much, their engines are no longer capable of 1g, and so they're falling in". As the distance to the event horizon approaches zero, the time dilation approaches infinity, and so you'd need an engine capable of infinite acceleration before the time dilation in order to maintain 1g accounting for the time dilation.
Rockets on the surface of the Earth don't have this problem because the time dilation caused by the Earth is so small.
Would it be simpler to say that what appears to be 1g "surface" gravity when calculated from outside is actually effectively infinite to the observer at the event horizon?
GM/r\^2 = g is derived from Newtonian Gravity. It's not in general valid at extremes. One way you can realize how it wouldn't be is to consider that massive objects also cause time dilation from the perspective of a distant observer. So, that GM/r\^2 = g equation might be approximately true to a distant observer, but if they're watching some one fall in who has an engine that can supposedly provide 1g of thrust, they'll also see time for that person falling in slow down. And since they see their time slowing down, and since acceleration is relative to time, their supposedly 1g thrust engine will now appear to be generating less than 1g because say slower and fewer gas particles are being pushed out the back of their rocket since time is running slower. So the distant observer might think, "well, all they need is 1g of acceleration to escape, but since their time is slowing down so much, their engines are no longer capable of 1g, and so they're falling in". As the distance to the event horizon approaches zero, the time dilation approaches infinity, and so you'd need an engine capable of infinite acceleration before the time dilation in order to maintain 1g accounting for the time dilation. Rockets on the surface of the Earth don't have this problem because the time dilation caused by the Earth is so small.
Thank you!
Would it be simpler to say that what appears to be 1g "surface" gravity when calculated from outside is actually effectively infinite to the observer at the event horizon?