I would like some help designing a budget tower speaker system

Well, with your size room, I would suggest no smaller than a 12” woofer. Often you can buy a completed unit for the same as the sum of the parts. Then you balance your requirements for the rest. For example if you like the high end consider a 4-5” FR. If you don’t need it as much, or if you want the add tweeters at a later date, consider 5-8“ FR. The satellite boxes you will want to build yourself as this might not be something readily available on the market.
The idea is to only do what you have to, to achieve what you want. KISS. No need to involve elements that turn into a complete headache when you get in too deep.
 
A pro woofer of typically 12" or 15" plus a waveguided tweeter.

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I do not use an equalizer.
Would a tweeter like this be a better match? This fall off around 500 hz and I'm finding large, 12-15", drivers that fall off around nearly the same point. Might not even need a crossover as they are pretty well matched
 
The situation may call for a modular construction. Build the D'Appolito front firing arrangement first. Then build the bass section. Stack the two. Arrange it so you can "stack" a separate rear firing module and top firing module (if you still want them).

Unsure why this isnt done more often. One would think a box "with the best sounding midrange" and other with "the best sounding tweeter" - etc - along of course with other boxes that have "other" midrange and tweeter units would be popular among DIY.

All designed to a decently engineered stackable form factor, where the sections of reproduction of the classic Woof-mid-tweet or Help-FR spectrum bands could be exchanged in a few minutes.

Like a ship of theseus speaker.
 
+1 to @AllenB's suggestions. Look at Pi Speakers kits. For the money, you could get a seven Pi corner speaker and a 3 pi subwoofer and have money left over. Alternatively you could combine the 3-pi sub and 3 pi into a single loudspeaker or forego the sub and use a pair of 3-pi, etc. Response curves are on the small graph icons on the specs and prices page.
 
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The situation may call for a modular construction. Build the D'Appolito front firing arrangement first. Then build the bass section. Stack the two. Arrange it so you can "stack" a separate rear firing module and top firing module (if you still want them).

Unsure why this isnt done more often. One would think a box "with the best sounding midrange" and other with "the best sounding tweeter" - etc - along of course with other boxes that have "other" midrange and tweeter units would be popular among DIY.

All designed to a decently engineered stackable form factor, where the sections of reproduction of the classic Woof-mid-tweet or Help-FR spectrum bands could be exchanged in a few minutes.

Like a ship of theseus speaker.
I also wondered why this isn't done more often. I feel like it could all be in the same tower with them just seperated by some pieces of MDF. If resonance with the connected boxes is an issue then could also sandwich acoustic board between the boxes in the tower between the two pieces of MDF