"given the extraordinarily low rate of mutations"
"low" is a relative term. In the case of corn, it's roughly 5.1 * 10^-5. Which sounds pretty low until you consider that each stalk will have 1-2 ears, and each ear has ~1000 kernels. And roughly 30,000 stalks per acre. Which means you're looking at dozens of mutations per acre. Yeah, processing that many kernels is going to be a royal PITA, but that's why we have undergrads ;) Even if you're just looking at grown plants, you're likely to see a mutation once every few acres.
"You are also ignoring the fact that in the time it takes for a new gene to be bred into a large portion of a species"
It is actually exactly the same either way. There isn't a massive assembly line of seed modification that alters 1 seed at a time. A select number of seeds are modified and breed in order to create the homozygous parents of the hybrid. As long as those seeds do not cross breed, you simply continue breeding that specified plant until you have sufficient quantities to sell (of both parents combined). It doesn't matter if the first seeds are from genetic manipulation or selective breeding.
"Contrast that with the ability to, say, insert bacterial genes into corn - something only possible through genetic modification"
I disagree with this assertion. It is possible through the sheer randomness of life for this same genetic change to occur. Again, I point to the existence of Roundup resistant weeds. They have by random mutation and environment changes developed similar genetic traits that closely match the intentional changes we made to the corn.
We could take the exact same approach to corn. Take non-Roundup ready corn, spray it with a small amount of Roundup, the more susceptible plants will die, the less susceptible plants will propagate. Next cycle, repeat the process, upping the dosage slightly. You will over a matter of a few generations wind up with a corn plant that is Roundup resistant. The problem though, is that you have no idea what other genetic traits have been propagated. Your corn may now have a longer growing period, may taste bad, may not be as disease resistant, etc... so you will need to take your new "non-GMO" Roundup resistant corn, and breed it back into plants that have your other desired traits. So after another pile of generations you'll wind up with a plant that is Roundup resistant, tastes good, grows well in your climate, and is resistant to drought/disease/etc...
Or instead of making it your life's goal of breeding corn, we can use technology to see what is the genetic difference between various Roundup resistant corn plants and non-resistant plants that have other traits we want. And in a period of just a few years you can have that same plant that would otherwise take a good bit of your life to breed.
At the end of the day, they are the exact same plant (genetically speaking). The only difference is whether you used science from the 1800's vs the 2000's.
There are risks to GMOs, specifically the homogenization of crops. For example, what we call a Banana today is not what our parents called a Banana decades ago. The banana propagates as a clone, exact genetic copies of itself. So if a disease or climate change occurs that effects 1 plant, it effect ALL plants. This is a risk with GMO as we wind up with far less genetic diversity by planting only the highest yielding seed lines. A single blight could wipe out masses of corn fields. But GMO doesn't cause such weaknesses, poor planning does. It takes all kinds, be it GMO, selective breeding, heirlooms, etc... bio diversity is a good thing.
-Rick