which way is more correct and methodologically appropriate?
Nothing is more "correct" or less correct, but there's certainly advantages and disadvantages. MCSCF calculations such as CASSCF calculations, can take a bit more mental energy if you want to make sure your active space is appropriate, but on the other hand, the result could be much more reliable than a black-box approach using DFT.
There is a third way which is perhaps better though. My default procedure for testing multiconfigurationality is to use FCIQMC: We can put a population of walkers on the Hartree-Fock determinant and then let the simulation run, and within seconds that population of walkers will spread to the other dominant configurations. We can then stop the calculation and see the populations on each determinant, and then decide things in the same way that you were suggesting, but we never needed to define an active space. The procedure is completely black-box. Another way to do this is using the HCI (heat-bath configuration interaction) approach instead of FCIQMC, or HCI can even be used as the excitation generator for the FCIQMC calculation.
However, you might not have an FCIQMC or HCI code installed yet, and maybe you want to use software with which you're already comfortable and familiar. In that case you can just do a coupled cluster calculation and look at the T1 diagnostic. First of all, if it's unusually difficult to converge the coupled cluster calculation, it can be because the system is multiconfigurational. If it's not too hard to converge the calculation but the T1 diagnostic is large, it is a sign that the system is not well represented by the dominant single configuration. A guideline is given in this PDF lecture where it is suggested that the T1 diagnostic should be <0.02 for closed shell systems, <0.03 for radicals, and anything >0.06 is a "catastrophe" for single-configurational methods like coupled cluster. The T1 diagnostic is usually written after a black-box coupled cluster calculation, and if it's not, you can instead look at the cluster amplitudes and decide multiconfigurationality in the same way you were considering to do it with an MCSCF calculation (the crucial difference being that you don't have to choose an active space for standard coupled cluster calculations, and the process is usually quite fast and black-box unless the molecule is too big, but if it's too big then your MCSCF calculation won't be very easy either).