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Chiral effective field theory ($χ$EFT), as originally proposed by Weinberg, promises a theoretical connection between low-energy nuclear interactions and quantum chromodynamics (QCD). However, the important property of renormalization-group (RG) invariance is not fulfilled in current implementations and its consequences for predicting atomic nuclei beyond two- and three-nucleon systems has remained unknown. In this work we present a first and systematic study of recent RG-invariant formulations of $χ$EFT and their predictions for the binding energies and other observables of selected nuclear systems with mass-numbers up to $A =16$. Specifically, we have carried out ab initio no-core shell-model and coupled cluster calculations of the ground-state energy of $^3$H, $^{3,4}$He, $^{6}$Li, and $^{16}$O using several recent power-counting (PC) schemes at leading order (LO) and next-to-leading order (NLO), where the subleading interactions are treated in perturbation theory. Our calculations indicate that RG-invariant and realistic predictions can be obtained for nuclei with mass number $A \leq 4$. We find, however, that $^{16}$O is either unbound with respect to the four $α$-particle threshold, or deformed, or both. Similarly, we find that the $^{6}$Li ground-state resides above the $α$-deuteron separation threshold. These results are in stark contrast with experimental data and point to either necessary fine-tuning of all relevant counterterms, or that current state-of-the-art RG-invariant PC schemes at LO in $χ$EFT lack necessary diagrams -- such as three-nucleon forces -- to realistically describe nuclei with mass number $A>4$.