Harbeth 40.3's. Should I buy them?

Before these 40.2 Anniversaries I have now, I had two different pairs of Daedalus Audio speakers (Ulysses, then Argos) Lou is the best! They are natural and organic sounding and not to just a percentage degree like some speakers. They truly are that. As well as incredible tonal realism. And boy can they ‘rock’ but with total clarity and incredible realism. I’d still have them but my mentioned sensitivities couldnt jive with the second angled tweeter that my weird sensitive ears couldn’t escape. I bet your wife would love the Daedalus sound. 
 

@decooney’s question and commentary points out an enduring quandary with integral floorstanders that are meant to run truly full range, versus using outboard subwoofers

we know how truly deep bass sound waves work within a listening room... our rooms have bass nodes/nulls, depending on their specific nature, and while a full range speaker like a spatial x3, a big harbeth, a big vandy, a big legacy, and so on... is designed to deliver real bass response down to 20-25 hz, say... well, that response will necessarily emanate from where the stereo speakers are placed in room ... this in effect forces the hand of the listener in setting up... where said speakers may image the best (based on their treble/midrange/midbass dispersion) may not be where the deep bass should best be delivered into the room to overcome bass nodes and nulls

such is the additional benefit of separately placed subs or sub ’swarms’, and why even ’full range’ speakers can benefit from being coupled with one or more subwoofers

Regarding the X3's, I shouldn't be running subs according to the manufacturer of my integrated. I currently use RCA's spades at one end and resistors on the other inside the RCA jacks. I do not have preouts on my integrated.  But I'm told that's a bad idea for many reasons. I need to stop that practice.

The foundation is really important to the music genres we enjoy. The reality starts to set in when I look at the 'big picture'. I picture a speaker with adjustable bass, woofers, preferably, almost certainly self-powered so that my tube integrated only see's the mids and tweets. I know by experience too that my wife likes a very dynamic picture. I do too. The speaker has to play Radiohead, Porcupine Tree, Shiny Toy guns, as well as female vocals, electronica, jazz, blues, and classical if it can pull it off. My experience with classical is that only big speakers with lots of drivers can pull off the scale of classical. Well, maybe horns too. I know most will disagree but that's been my experience. 

My room is pretty good. 15' x 19' dedicated with GIK bass traps and acoustic panels at the front, 1st order reflections taken care of, Cement slab, thick carpet, rockwool on one wall,  LCR panelboard but a dead back wall as it's nothing but shelving holding vinyl. So no 'live end - dead end'. I don't hear bass issues with any of the speakers in that room not that they don't exist of course.

Swarms are the best way to get a good bass response, fully agree, but I don't have that option. Another reason why I keep thinking that the Legacy Audio Focus XD's seems to check a lot of boxes.

Regarding the X3's, I shouldn't be running subs according to the manufacturer of my integrated.

Start with speakers, then the amplifier. If this is the truly the case according to your amp manufacturer, imo the amplifier is a limiting factor - given all lf the different speakers you've tried and still plan to. Same issue, you'll want to hear the Legacy with your integrated.  Not their amps.

A close colleague went and demo'd them, he walked out being led to believe he needed to buy much higher power SS amps to get the most out of the Focus XDs and a few others. Was told his 60w mono tube amps wouldn't cut it. Bailed out.  

With a really good preamp, with two sets of pre-outs, the right mono amps, the right subs, guarantee without a doubt those X3s could be made to sing. :) my .02 cents. 

You must be talking about adding additional subs to the onboard subs of the x3? Not sure why that would be needed, the bass performance of this speaker is the best thing about it.