You never stated your conditions
Or maybe you didn't read carefully my posts?
I have never stated anything to the contrary, but time delay and phase are not strictly the same thing.
Here you go again : with circular movement time and phase are DIRECTLY related via frequency and speed of the wave. It is absolutely irrelevant which one you want to use as they are interchangeable.
Ok, so explain why speakers that are not time aligned between woofer and tweeter (the majority) sound (generalising) OK?
This one is an interesting question and from what I have seen many folks are not clear with it. Crossover specialist would do that much better but let me give it a try.
First of all, with decently sounding 2-way speakers woofer and tweeter are absolutely phase aligned at XO point. Let's see how it works with my speakers:
Blue line is phase response of my uncorrected speaker. XO is at 1800Hz, LR4 24db/octave. If I didn't tell you that you never would have guessed as phase response is very smooth in that region which means at 1800Hz they are phase aligned. And they should be otherwise there would be a horror in frequency response around XO point.
Let's have a look at step response, again blue is uncorrected. It looks typical, doesn't it?
Red lines are phase and step response after phase of the passive LR4 crossover at 1800Hz has been corrected using this phase filter:
Here's another view of the same filter so you get a better idea what it does:
As a conclusion: although woofer and tweeter are phase aligned at the XO point passive LR4 XO shifts the filter all over the audible range and that is what you are seeing while looking at uncorrected step response. But if you correct it you will end up with a very decent phase and step response. That of course still depends on the quality of the drivers of your speaker as junk drivers/speakers cannot be EQ-ed whatever you do.
Different types of crossovers will of course have different phase signatures but all of the standard ones can be corrected and that is what automatic EQ tools like Acourate, Dirac etc are doing. When they mention they correct things in time domain they first correct phase response of passive XO and then they try to linearize phase response to get excess phase (difference between actaul and minimum phase response) as close to 0 while avoiding pre-ringing with step response as that causes audible artifacts which are much worse than high GD.
And that's about it, what can fit on one page, hope it helps!