coaxial vs fullrange

Yes, even a zen purist full range system should have an appropriate high pass filter that is exactly tuned to the bass reflex, transmission line, or horn frequency cutoff. Very few FR designs have that.

Unfiltered designs are flapping in the breeze and nobody notices. It’s FAR worse in analog systems with turntable rumble. An appropriate filter extends dynamic range by 5-10dB.
This!
I cannot emphasise this clearly enough. This also applies to conventional bass reflex designs, as long as they are not tuned so low that audio signals below them becomes less and less likely, e.g. with my four subwoofers with Eminence LAB 12 in the living room, whose fb is just over 20 Hz.

However, I also have two subwoofers in operation here with an fb of around 40 Hz. It's incredible how level stable they become when you give them a filter that doesn't increase the excursion below fb any more than above it; in this case it was an LR4 highpass filter at around 30 Hz.

Yes, it increases the group delay at the bottom, but firstly this is not particularly audible in this frequency range, and secondly there are also a few tricks to avoid increasing the group delay too much.
 
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@Sonce
you do not screw up the phase if you have the indirect tweeter doing nothing below 5khz, 18db filters can help to achieve this.
Just read here about it:
https://en.wikipedia.org/wiki/Precedence_effect
Making phase chaotic (with adding indirect tweeter to a fullrange) has nothing to do with the precedence effect. Precedence effect means that chaotic phase sound is coming apparently (subjectively) from the same fullrange, not from the tweeter - just that, nothing else. it will not maintain the linear phase which fullrange aficionados are raving about - мeasurements easily reveal that. Of course that phase "problem" will not be audible, just as the similar phase "problem" in 2-way loudspeakers will not be audible - ears are not so sensitive to phase problems in high frequencies.

The tweeter should have directivity properties, and if you choose beaming to the sides, up or behind the box in order not to interfere with the wave from the front it is not harming the direct wave.
Only if tweeter is in waveguide/horn.

However indirect tweeters do not need to be time aligned like in a two way where the tweeter beams with the main driver together to the listener.
Yes, but see above comments.