Purely theoretical question about idelal filters and infinite oscillations
Note: historically the Gibbs phenomenon was described over a continuous-time domain periodic square wave which was constructed from a finite number of its Fourier series coefficients, however, one can interchange the domains and observe the same behaviour on the frequency response of a finite length discrete-time filter with its impulse response samples designating the Fourier series coefficients.

This question is essentially asking the consequences of the unavoidable Gibbs phenomenon on the mathematical construction of an ideal brick-wall filter whose frequency domain definition apparently (and confusingly) does not involve the expected Gibbs ripples...?

The practical answer is that as the filter length approaches infinity the peak magnitude of the ripples does not reduce, but its duration shrinks to a single point at the edges of the passband. From Riemannian integration point of view, this will not have an effect on the filter's output as if there's no Gibbs phenomenon.

On any finite length approximation of an ideal brickwall filter, however, the ripples do exist, and will alter the actual output from that of an ideal filter's computation. The better solution to reduce those ripples is to make use of a window instead of increasing the filter impulse response truncation length.