My first thought after reading the initial post was to try different IRs, they are absolutely crucial in replicating any tone in the Axe. I've been on the Fractal train since 2015 and still have a few tube amps, none of which have been played since moving from the AX8 to the Axe-Fx III. The only difference for me has been playing an amp with a cab vs the Axe III with monitors. Tonally, there is no difference. Amp response, there is no difference. Feel, there is no difference. Air movement or filling the room with sound, there is a difference.
I realize that AITR is not part of the discussion here since you used a load box for the comparison. As Cliff pointed out, this changes the baseline of the test. Have you tried the Axe III in the effects loop of an amp going into the load box? It would be interesting to hear your thoughts after performing this test.
With all of the parameters in the Axe III, in combination with different IRs, it is very possible to 'close the gap' yourself should you desire to delve into the advanced parameters deeper.
I've said it before, I am very particular about my tone and will use whatever gives me what I need to achieve the tones I need. If an amp and pedalboard performed 'better' than the Axe III I would use it, hands down no question. The Axe Fx III provides everything I want or need without any compromise.