One may indeed find good reasons for such reasoning to be correct, although I believe they both (IR and step) should yield the same result.
Main problem, however, in this case was that the initial measurement of mains and sub that
@ernestcarl provided was obviously flawed by misconfigurations in main's DSP, miniDSP XO settings and XO and phase settings in sub (which cannot be bypassed by using LFE input as there is none).
Look for example at step response of the sub:
View attachment 77618
Not bad at first sight, right?
Now comes several "but"s..
Sub was supposed to be measured, together with right main speaker, using left main to provide common timing reference so that delay can be estimated. XO for the channel 4 driving the sub was supposed to be bypassed so that only XO set on sub is active - but it was not. XO at miniDSP on channel 4 was active, with XO on set to unknown parameter.
Phase on the sub was supposed to be set on 0 deg but it was set on 180 deg.
All this makes this measurement unusable.
Now let's look at the right channel step:
View attachment 77619
Ok, this one doesn't even look right on the first sight. It should be similar to the sub but instead it's a picture of sheer horror.
It was supposed to be made with XO HP setting on miniDSP channel 2 at 80Hz and delay of 0ms. PEQs on miniDSP should be disabled and all internal DSP settings on the main speaker should be disabled.
But that was clearly not the case, otherwise it would look much more similar to the step of the SW.
So, problem here is not in the methodology (using IR vs using step) but more into making clean precise measurements which would yield reliable and usefull result.
To answer your question, I believe both, IR and step, can be used as both should have single dominant clean peak which can be then compared to the similar peak of the other speaker to precisely determine the delay.
The only way out from this mess that I can see is get the settings right and repeat the measurements.