Key heuristic
For a Brownian increment,
$$ B_{t_{i+1}}-B_{t_i}\sim N(0,\Delta t), $$so
$$ \mathbb E[(B_{t_{i+1}}-B_{t_i})^2]=\Delta t. $$Why this matters
For smooth paths, squared increments are of order $(\Delta t)^2$, so quadratic terms disappear in the limit.
For Brownian motion, squared increments are of order $\Delta t$, so summing them over roughly $T/\Delta t$ intervals gives an $O(1)$ contribution.
Consequence
The second-order term does not vanish, and this is the source of the extra correction term in Ito calculus.