It's close, not trying to be perfect and not striving to be perfect, I'd think the former is a foregone conclusion of unattainable goal. However the later seems appropriate more as an attitude towards work. Kind of subtle "to try to be something" this sort of carries notes of fabrication and possibly hubris? Alternatively, "to strive to be something" carries a degree of humility of knowing you may not attain it, but recognizing it's the process of convergence that matters not the ultimate end state. I may be reading too much into this but those are the thoughts that cone to mind.
For one, it's a reference to the Constitution: "We the People of the United States, in Order to form a more perfect Union, establish Justice, ensure domestic Tranquility, provide for the common defence, promote the general Welfare, and secure the Blessings of Liberty to ourselves and our Posterity..."
Second, it's a much weaker statement. Compare "we are [imperfect], but that doesn’t mean we are striving to form a union that is perfect." with "we are [imperfect], but that doesn’t mean we aren't striving to form a union that is perfect." The former can be reshaped to we are imperfect, therefore we are trying to form a union that is imperfect---it leaves open the possibility that you wish to make the union worse. The latter becomes we are imperfect but we are trying to form a union that is perfect.
When you do something that is important to you, do you try to do the best you can, try to make it perfect? (Perhaps with the realization that it cannot be perfect.) Or do you think, "Eh, imperfect is good enough?"
"We the People of the United States, in Order to form a more perfect Union, establish Justice, insure domestic Tranquility, provide for the common defence,[note 1] promote the general Welfare, and secure the Blessings of Liberty to ourselves and our Posterity, do ordain and establish this Constitution for the United States of America."
To other answers here: the phrase in the poem "we're not trying to be perfect" seems contradicting the goal "to form a more perfect Union", unless you think that "let's be perfect" and "less be more perfect" are different enough. I think the road to perfection shares parts with the road to more perfection - often.
I think that it is intentional to underscore that "more perfect" != "perfect". The founders that wrote that document did not believe the Constitution was perfect, they knew it a part of process towards perfect, while it might imply perfect is an end goal it also doesn't necessarily say that.
Listening to her recitation at [1] she does appear to say "we are striving". It might be a simple slip, or she could be saying that the objective is purpose not perfection. I tend to think it's the latter.
Um,...?