1. "I believe that I am [qualified to be POTUS] because I have common sense."
Nothing grammatically wrong with this sentence, that I can see. The only thing worth noting is her premise that common sense constitutes a qualification for holding public office. And not just a qualification, but the first one worth mentioning.
2. "And I have, I believe, the values that are reflective of so many other American values."
Structurally, this sentence is okay, but what she means is anyone's guess. She has values that are reflective of other values? Her sentence seems to imply that she does not actually have any of those "other" values, but simply has values that reflect the others. We apparently are dealing with two sets of values here: (1) The values that Palin has (2) The values that are merely "reflective" of her values, but which she does not possess herself.
And all this is somehow supposed to relate to her qualifications to be president. All we've learned from it is that she's fit to be president because she has American values that reflect other American values.
3. "And I believe that what Americans are seeking is not the elitism, the kind of a spinelessness that perhaps is made up for that with some kind of elite Ivy League education and a fact resume that's based on anything but hard work and private sector, free enterprise principles."
This jigsaw puzzle of a sentence had me practically screaming the first time I saw it. Let's try to put it together.
Her first point is that Americans are not seeking "the elitism." What elitism? The comma that follows the word indicates she's about to define the concept. And she defines it as "the kind of spinelessness that perhaps is made up for that with some kind of elite Ivy League education...." Okay, stop there for a moment. Never mind the awkward wording ("made up for that with"). Her definition of "elitism" is spinelessness that is "perhaps" made up for with elite education.
By using the word "perhaps," she suggests that elitism could simply mean spinelessness, without any of the other qualifying properties. So elitism in the purest sense means lacking a spine. I guess the Cowardly Lion could be called the Elitist Lion. I guess snails, insects, and other invertebrates could be called Elitist Organisms.
We will have to ignore the word "perhaps" to get any further with this sentence.
The final quality she mentions in her definition of elitism is "a fact resume that's based on anything but hard work and private sector, free enterprise principles."
What in the world is a "fact resume"? I'm not even going to attempt to figure it out.
So, putting all this together, she argues that Americans aren't seeking an elitist, and she defines an elitist as a spineless person who perhaps makes up for his spinelessness with an elite education and a resume free of hard work and private-sector principles. Doesn't that sound like every Ivy Leaguer you know?
4. "Americans could be seeking something like that in positive change in their leadership."
After that last nightmare of a sentence, this one is almost a piece of cake. But remember, she previously stated what Americans are not seeking. She never explained what they actually are seeking. So what are we now to make of her insistence that Americans are seeking "something like that"? Something like what? Does she mean anyone who does not fit the previous description? Okay, it has to be a person who possesses vertebrae. And it can't be someone with an Ivy League education--or at least it can't be someone who uses their Ivy League education to make up for the first flaw. And it can't be someone who has a resume without private-sector principles.
So, in sum, what Americans are seeking is a non-Ivy League vertebrate who doesn't have a resume lacking in private-sector principles. Got that.
5. "I'm not saying that has to be me."
Of course you aren't. Who would have suggested such a thing? After all, every qualification you've mentioned--common sense, values that reflect other values, backbone, lack of Ivy League education, absence of a resume that doesn't have private-sector principles--would describe millions of Americans.
What you seem to have neglected to explain is how you managed to get on the short list.