In the presence of a sufficient amount of target material, gamma rays can be used as a tracer in the search for sources of Galactic cosmic rays (CRs). Here we present deep observations of the Galactic Center (GC) region with the MAGIC telescopes and use them to infer the underlying CR distribution and to study the alleged PeV proton accelerator (PeVatron) at the center of our Galaxy. We used data from 100hr observations of the GC region conducted with the MAGIC telescopes over five years (from 2012 to 2017). Those were collected at high zenith angles (58-70deg), leading to a larger energy threshold, but also an increased effective collection area compared to low zenith observations. Using recently developed software tools, we derived the instrument response and background models required for extracting the diffuse emission in the region. We used existing measurements of the gas distribution in the GC region to derive the underlying distribution of CRs. We present a discussion of the associated biases and limitations of such an approach. We obtain a significant detection for all four model components used to fit our data (Sgr A*, "Arc", G0.9+0.1, and an extended component for the Galactic Ridge). We observe no significant difference between the gamma-ray spectra of the immediate GC surroundings, which we model as a point source (Sgr A*) and the Galactic Ridge. The latter can be described as a power-law with index 2 and an exponential cut-off at around 20TeV with the significance of the cut-off being only 2{sigma}. The derived cosmic-ray profile hints to a peak at the GC position and with a measured profile index of 1.2+/-0.3 is consistent with the 1/r radial distance scaling law, which supports the hypothesis of a CR accelerator at the GC. We argue that the measurements of this profile are presently limited by our knowledge of the gas distribution in the GC vicinity.