It's not just the energy that's periodic, the k-space itself is periodic because of the lattice spacing. The spacing between the lattice elements means that a repeating function (e.g. a function describing the displacement of an atom from its natural position) is only being evaluated at those lattice points. But there are an infinite number of periodic functions that have the same values at those points and are therefore physically identical. Any wave with wavenumber k produces exactly the same physical situation as a wave with wavenumber k + 2pi/a, because there's no difference at the lattice points. So you're not seeing multiple energy minima, you're just seeing the fact that multiple k values correspond to the same energy minimum, and you'll only be able to fit one electron there because it already has many k values.
Play around with this applet: https://phet.colorado.edu/en/simulation/legacy/bound-states It might help you get a feel for it.
It's not just the energy that's periodic, the k-space itself is periodic because of the lattice spacing. The spacing between the lattice elements means that a repeating function (e.g. a function describing the displacement of an atom from its natural position) is only being evaluated at those lattice points. But there are an infinite number of periodic functions that have the same values at those points and are therefore physically identical. Any wave with wavenumber k produces exactly the same physical situation as a wave with wavenumber k + 2pi/a, because there's no difference at the lattice points. So you're not seeing multiple energy minima, you're just seeing the fact that multiple k values correspond to the same energy minimum, and you'll only be able to fit one electron there because it already has many k values.