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Massive stars can explode in powerful supernovae (SNe) forming neutron stars but they may also collapse directly into black holes (BHs). Understanding and predicting their final fate is increasingly important, e.g, in the context of gravitational-wave astronomy. The interior mixing of stars in general and convective boundary mixing remain some of the largest uncertainties in their evolution. Here, we investigate the influence of convective boundary mixing on the pre-SN structure and explosion properties of massive stars. Using the 1D stellar evolution code Mesa, we model single, non-rotating stars of solar metallicity with initial masses of $5-70\mathrm{M_\odot}$ and convective core step-overshooting of $0.05-0.50H_\mathrm{P}$. Stars are evolved until the onset of iron core collapse, and the pre-SN models are exploded using a parametric, semi-analytic SN code. We use the compactness parameter to describe the interior structure of stars at core collapse. Larger convective core overshooting shifts the location of the compactness peak by $1-2\mathrm{M_\odot}$ to higher $M_\mathrm{CO}$. As the luminosity of the pre-SN progenitor is determined by $M_\mathrm{CO}$, we predict BH formation for progenitors with luminosities $5.35<\log(L/\mathrm{L_\odot})<5.50$ and $\log(L/\mathrm{L_\odot})>5.80$. The luminosity range of BH formation agrees well with the observed luminosity of the red supergiant star N6946BH1 that disappeared without a bright SN and likely collapsed into a BH. While some of our models in the luminosity range $\log(L/\mathrm{L_\odot})=5.1-5.5$ indeed collapse to form BHs, this does not fully explain the lack of observed SN~IIP progenitors at these luminosities, ie the missing red-supergiant problem. Convective core overshooting affects the BH masses, the pre-SN location of stars in the Hertzsprung-Russell diagram, the plateau luminosity and duration of SN~IIP lightcurves.[Abridged], Comment: Accepted for publication in Astronomy & Astrophysics: 23 pages, 14 figures