The $n$-point correlation functions of QCD, which define the theory, are computed by performing functional derivatives on $Z_{QCD}[J]$, the generating functional of QCD,
$$\frac{\delta^nZ_{QCD}[J]}{\delta J(x_1)...\delta J(x_n)},$$
where $J$ are the different field sources that are to be set to zero. In a perturbative approach, we can expand on the weak gluon coupling constant, but doing so will require taking into account the presence of the loop integrals that, generally, diverge and one has to deal with these singularities that are a consequence of allowing the integrations to run over every possible energy at every point in space-time.
The first step is regularizing the theory, where a regulator is introduced in the loop integrals such that they become convergent. Doing this then allows one to redefine the gluon coupling constant, quarks masses and fields strength, that appear in the generation functional of QCD, in such a way that they cancel these infinities. This redefinition is called renormalization and by doing so we can compute the finite renormalized correlation functions.
From this perturbative approach, we also see that the coupling and mass will be dependent on the energy scale, such that the correlation functions are independent of this scale.
Now, starting again from the step of computing the correlation functions by functional derivatives on $Z_{QCD}$ but now from a non-perturbative approach (and forgetting everything about the perturbative approach), I understand that there isn't a way to do this computation and one has to use other non-perturbative methods such as lattice QCD. But where does the need to regularize and renormalize QCD come in? Everywhere I have looked for answers they always rely on the perturbative approach, using arguments such as problematic divergencies or scale-dependent mass and coupling, and I don't see how they are present in the non-perturbative approach.
The only 2 leads that in my eyes could lead to problems are the fact that the coupling and mass are bare parameters with no physical value and that the theory is defined as allowing every possible energy at every point in space-time, but I don't understand where these apparent pathologies come in.