The U.S. federal government is strong because it was designed to be strong. It was designed to be strong because the nation had already tried a weak national government under the Articles of Confederation, and it was an abject failure.
Not even Scalia would object to the notion that the federal government is strong. In fact he regularly rules in favor of a strong federal government, including against Larry Lessig in Eldred v Ashcroft.
I wonder if the educational system is responsible for this idea that the Constitution was designed to have a "weak government" that wouldn't be "worth taking over." Growing up in Virginia, I definitely had a "states are so amazing" slant to my education (this is the same state that has a giant statute of a traitor in its capital). I don't know what people who grew up outside the south were told.
The fact is that the Founders didn't really agree on how strong the Federal government should be, but they all agreed it should be stronger than the weak national government that existed under the Articles of Confederation. And the Federalists wanted a very robust national government indeed (and remember, the reference point of the British Parliament, which has nearly unlimited power, was the framing in this situation), while others wanted to preserve the nearly unlimited power of the states. Nobody wanted a "weak" government--it was a battle about who should hold the power.
And it was a battle that the Federalists won in the first several Congresses and during the Marshall era, and finally consolidated at the point of a sword during the civil war.
Not even Scalia would object to the notion that the federal government is strong. In fact he regularly rules in favor of a strong federal government, including against Larry Lessig in Eldred v Ashcroft.