The underlying motivation for linearizing an equation of motion is that you are looking for small deviations to a known solution of the full equations of motion. The approximation is valid so long as these deviations remain small. If the deviations do not remain small, then the approximation breaks down.
A linear term in the Lagrangian corresponds to a constant term in the equations of motion. From your intuition from classical particle mechanics, you know this constant term would correspond to a constant force, leading to a constant acceleration. A perturbation experiencing a constant acceleration will inevitably become large, meaning that the expansion is breaking down.
The underlying reason you are finding this instability is that you are not expanding around a legitimate solution of the full equations of motion. If you were, the linear term would vanish. This follows because the principle of least action states that all first order variations of the action vanish when the Euler-Lagrange equations are satisfied.
In the special case of a potential, then a constant value of the field sitting at the minimum of the potential is a valid solution of the equations of motion, but a constant value of the field sitting at some random spot on the potential that has a non-vanishing derivative is not a solution -- the equation of motion will make the field roll down the potential.
