As far as I know the "naturalness" problem comes from trying to understand some physical constants at lower energies from an assumed high energy model. So if I have some Lagrangian at low energies with a small physical constant , I could consider a possible ultra violet completion. In the high energy Lagrangian I could do a renormalization group flow to the lower energy Lagrangian. It turns out that in some theories constants that are small in the lower energy Lagrangian get multiplied by positive powers the cut off after the RG flow and so these constants depend sensitively on the cutoff. The question then becomes if we observe them to be small there must be some mechanism to cancel the ultraviolet corrections coming from the high energy Lagrangian. If you can't find such a mechanism then you have to tune these constants finely so that everything works out.
On the other hand there are constants that get multiplied by negative powers of the cutoff and so do not matter at low enough energies. So the issue is not simply about constants being " too small or too big". For example the mass of the electron is smaller than the Higgs particle but no one worries about the mass of the electron.
Other constants could be determined by a symmetry. So for example, the fact that there is a limit on the speed information can be transferred is a result of Lorentz symmetry. This speed is the speed of light in vacuum; it may be a big number but it is determined by a symmetry.
There is a separate question of whether one can calculate all physical constants in terms of mathematical constants. This is a separate issue but no unrelated. It is not unrelated because if all constants are determined by mathematical constants then it means that "naturalness problem" is ultimately a fake. We are just not smart enough or know enough to figure out the right theory.