Intuition behind dependence on $m^2$ in classical scalar field theory Lagrangian

I think you're confusing two different masses: the mass of the springs and the mass of the field quanta.

Start with coupled masses-on-springs, of mass $M=1$ and spring constant $k$. Let's imagine that the springs are located on a 2d lattice, and they oscillate only in a 3rd direction.

Quantize and take the continuum limit, and you get a quantum field theory, which describes the values of a field which lives on $\mathbb{R}^2$. This QFT has quanta, which have mass $m = \sqrt{k}$. These particles are not_ the original masses-on-spring though; they're collective oscillations of the springs. Their mass governs how fast they move around in $\mathbb{R}^2$, not how fast they move transverse to the 2d plane.

Note that you don't have to (and shouldn't!) think of the original springs as oscillating in the same space as the lattice points. The springs are part of the example because we wanted a system described by a number at each point in space.