First, I'm using the common definition of battleground states, which is states that are currently considered winnable by both sides. That doesn't include New Hampshire, or any of the smallest states.
Second, arguable means you can make a good argument for something, so I think you just proved that it's arguable. It is not a slam dunk.
The only advantage of less populated states is that they get two "free" electors regardless of their population. This effect is strongest in small states, where it helps both parties equally.
Looking at all the states, the maximum advantage to a presidential candidate is the difference between the number of states they won times two. For example, if both candidates win 25 states, then the two "free" electors per state will cancel out and the electoral college will be determined solely by the number of representatives in the states that each side wins. Or to put it differently, if the Constitution were "fixed" so that electors were strictly awarded by population, then the winner would never change in a 25 to 25 split.
Of course, if one candidates wins 26 states and the other wins 24 states, then the first candidate could potentially get four "unfair" electors by winning more small states. But historically, the electoral college is won by much larger margins. The only modern candidate who might have won if the Constitution were "fixed" would have been Gore, and that was a highly unusual election. Otherwise, the small state advantage hasn't made a significant difference in our lifetime.
