I almost certainly cannot go out and listen to all of these so I need to narrow my options before I go out and start listening. Starting with an integrated amp seems like a reasonable thing to do.Sorry but I disagree. Starting with the speakers sounds like the most reasonable thing to do. After you settle upon the speakers that sound best to you, then select the amp that will properly drive those speakers to whatever sound levels you're comfortable listening at without any fear of clipping.
So suggest you start auditioning speakers immediately, even if they are not the ID brands you list above. BTW, why those? Have you heard them? The Airmotives are active speakers meaning you don't need an amp. The Direct Acoustics are 6 ohm nominal, 4 ohm minimum with no sensitivity figures published. From the looks of these, neither is the ideal match for the amps you mention.
So before you spend a dime, suggest you google the two big speakers specs, impedance and sensitivity. Learn and understand them and how they relate to an amp. Then go out and audition as many speakers as you can paying attention to these specs, their driver compliment and how they ultimately sound to you. You'll be on your way to spending your 2 grand far more wisely.