In the case of a simple pendulum, it is physically intuitive that the time-period should increase with the increase in l (distance traveled over an oscillation is increasing linearly with l). In the same sense what is the intuitive physical explanation of this apparently counter-intuitive behavior?

In my opinion, the behaviour of a simple pendulum is not so intuitive. I guess if a common visitor of the Paris Pantheon is asked about the period of the Foucault's pendulum, it would be no surprise if some answers put the mass as a variable for example, and miss the role of the length.

But, we can have an "educated" intuition of the rigid body oscillations: as the moment of inertia is $\alpha mr^2$, where $\alpha$ is some constant, increasing $l$ means decreasing the length parameter ($\frac{r^2}{l}$) inside the square root. The period is proportional to the square root of the "length" so to speak. And the mass cancels.

So its behavior is intuitive in this meaning.

In the case of a simple pendulum, it is physically intuitive that the time-period should increase with the increase in l (distance traveled over an oscillation is increasing linearly with l). In the same sense what is the intuitive physical explanation of this apparently counter-intuitive behavior?

In my opinion, the behaviour of a simple pendulum is not so intuitive. I guess if a common visitor of the Paris Pantheon is asked about the period of the Foucault's pendulum, it would be no surprise if some answers put the mass as a variable for example, and miss the role of the length.

But, we can have an "educated" intuition of the rigid body oscillations: as the moment of inertia is $\alpha mr^2$, where $\alpha$ is some constant, increasing $l$ means decreasing the length parameter ($\frac{r^2}{l}$) inside the square root. The period is proportional to the square root of the "length" so to speak, if l is small compared to the other term. And the mass cancels.

So its behavior is intuitive in this meaning.