Horn speakers , high efficiency but not “shouty”

Much of the cherished audiophile recordings were mastered using horns the live music you all use as a reference played back on horns. Thinking a wee box or thin panel is even close to what a proper horn can do is just ignorant biased delusion. 

No proper horn is shouty. Cheap undersized poorly designed horns over a ported box can have issues since midrange is so much more efficient most compromised to make them smaller cheaper. A proper horn is fully front horn loaded to at least 100hz not many like this exist that audiophiles would buy so they opt for small poorly designed horns then go online saying that all horns shout or some such BS.

TAD horn is designed for a TAD driver its phase plug improves loading for such why it exists the problem is folks using the TAD horn for non TAD drivers.

I have built over 100 full-range driver systems. What you are describing is a characteristic that they can have if cabinets are too lively.  All lack extension on top and bottom =unless giant. I would consider a bass horn system or multiple large subwoofers and adding a tweeter like a Fostex t900a. 300b amplifiers while lovely things some can sound thick and a bit sloppy and can cause resonance issues with some BLH. When I build full-range systems they are very large and the cabinet is very much overbuilt to reduce any coloration.