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2. The Scale of the Starman & Mega Mushroom
If the localized force is insufficient proof, we turn to temporary power-ups whose functions are inherently destructive:

The Mega Mushroom: This transformation grants Mario transient, reality-bending mass and momentum. While Mega, he runs through obstacles—including castles, skyscrapers (in SMW), and giant enemies—displacing them entirely. Running through a mountain would not stop him; it would merely create a geographically significant tunnel or, more likely, displace the entire mass due to the instantaneous, physics-defying momentum Mario achieves.
The Super Star: Temporary invincibility is not merely protection; it is a declaration of localized temporal and physical immunity. Anything Mario touches while under the Star’s effect is instantly annihilated or negated (Goombas, Koopas, even flames and energy attacks). If Mario can touch a mountain while invincible, the mountain’s structural integrity must, by the established rules of the Super Star, be negated.
3. Lifting Bowser: The Planetary Anchor
In numerous titles (notably Super Mario 64), Mario grabs and swings Bowser—a beast whose size, weight, and density are enormous, capable of surviving orbital drops and volcanic immersion. The force required to centrifuge a being of Bowser’s mass—often performed in the vacuum of outer space or high-gravity environments—equates to a megaton-level feat of pure kinetic manipulation.

Conclusion on the Mountain: Mario is a highly efficient hero. If a mountain impedes his path, he has three options: go over it, go through a pipe around it, or use a feather to fly past it. Destruction is the tool of the brute, not the artisan. The mountain remains because Mario has never deemed its removal to be the most expedient route to the princess.