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Background: The properties of very neutron rich nuclear systems are largely determined by the density dependence of the nuclear symmetry energy. The Korea-IBS-Daegu-SKKU (KIDS) framework for the nuclear equation of state (EoS) and energy density functional (EDF) offers the possibility to explore the symmetry-energy parameters such as $J$ (value at saturation density), $L$ (slope at saturation), ${K}_{\mathrm{sym}}$ (curvature at saturation), and higher-order ones independently of each other and independently of assumptions about the in-medium effective mass, as previously shown in the cases of closed-shell nuclei and neutron-star properties.Purpose: We examine the performance of EoSs with different symmetry energy parameters on properties of nuclei and observations of neutron stars and gravitational waves in an effort to constrain in particular $L$ and ${K}_{\mathrm{sym}}$ or the droplet-model counterpart ${K}_{\ensuremath{\tau}}$.Method: Assuming a standard EoS for symmetric nuclear matter, we explore several points on the hyperplane of $(J,L,{K}_{\mathrm{sym}}$ or ${K}_{\ensuremath{\tau}})$ values. For each point, the corresponding KIDS functional parameters and a pairing parameter are obtained for applications in spherical even-even nuclei. This is the first application of KIDS energy density functionals with pairing correlations in a spherical Hartree-Fock-Bogoliubov computational code. The different EoSs are tested successively on the properties of closed-shell nuclei, along the Sn isotopic chain, and on astronomical observations, in a step-by-step procedure of elimination and correction.Results: A small regime of best-performing parameters is determined and correlations between symmetry-energy parameters are critically discussed. The results strongly suggest that ${K}_{\mathrm{sym}}$ is negative and no lower than $\ensuremath{-}200\phantom{\rule{4pt}{0ex}}\mathrm{MeV}$, that ${K}_{\ensuremath{\tau}}$ lies between roughly $\ensuremath{-}400$ and $\ensuremath{-}300\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$ and that $L$ lies between 40 and 65 MeV, with $L⪅55\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$ more likely. For the selected well-performing sets, corresponding predictions for the position of the neutron drip line and the neutron skin thickness of selected nuclei are reported. The results are only weakly affected by the choice of effective mass values. Parts of the drip line can be sensitive to the symmetry energy parameters.Conclusion: Using KIDS EoSs for unpolarized homogeneous matter at zero temperature and KIDS EDFs with pairing correlations in spherical symmetry we have explored the hyperplane of symmetry-energy parameters. Using both nuclear-structure data and astronomical observations as a testing ground, a narrow regime of well-performing parameters has been determined, free of nonphysical correlations and decoupled from constraints on the nucleon effective mass. The results underscore the role of ${K}_{\ensuremath{\tau}}$ and of precise astronomical observations. More-precise constraints are possible with precise fits to nuclear energies and, in the future, more-precise input from astronomical observations.