Well, Nike already answered the point about the variationality: even though methods like MP2, CCSD, and CCSD(T) are non-variational in that they may over- or underestimate the energy of the ground state (or excited states) of the Schrödinger equation, the energy reproduced by any given method typically does behave variationally with respect to the basis set. You can understand this in another way: while CCSD(T) doesn't even have a wave function, you can write a Lagrangian for it, which the solution minimizes. The one-electron basis set just affects the accuracy with which you're minimizing this functional.
Note that this is not restricted to ab initio calculations. DFT is a celebrated class of methods, but the absolute energies reproduced by density functional approximations (DFAs) don't mean anything. DFT is most certainly non-variational with respect to the true solution to the Schrödinger equation; still, there are many atomic-orbital basis sets that are optimized for the minimization of DFT energies.
I would still like to point out that absolute energies are for sure important for doing accurate benchmarks: the whole point of extrapolation schemes is that the error with respect to the basis set is monotonic, and the total energies approach the exact value from above.
If you're studying a new property / a new level of theory, you should always check the convergence with respect to the basis set. If the changes from XZ to (X+1)Z are small, this typically means that you have reached convergence to the basis set limit; your absolute energy should then also be close to the correct value. This does not always happen though: even though you can reach sub-microhartree level accuracy in the total energy for self-consistent field calculations for light atoms in Gaussian basis sets, for heavy atoms the absolute energy is only accurate to millihartree [J. Chem. Phys. 152, 134108 (2020)].
PS. An interesting exception can be found in relativistic methods. The Dirac equation permits two classes of solutions: the electronic solutions with positive energies, and the positronic solutions with negative energies (a.k.a. Fermi sea). In a given AO basis set with K basis functions, you will have K positive-energy solutions, and K negative-energy solutions. The energy is not determined as a minimization of the energy functional with respect to orbital rotations as in non-relativistic theory, but rather as a mini-max procedure where you minimize with respect to electronic rotations and maximize with respect to positronic rotations. Because of this, the variational principle no longer applies. You can see this effect e.g. in calculations with the Douglas-Kroll-Hess (DKH) and exact two-component (X2C) approaches.