This isn't going to be a super-elaborate Instructable, but I wanted to share it in case someone else was interested in doing such a thing but perhaps hung up on something I solved (like the trapdoor action).

TinkerCAD link

Adjust the dimensions to suit yourself, but I went with about 8.5" (214mm) tall and deep by about 4" (110mm) wide. Really, this was pretty arbitrary, so adjust it to your materials.

Supplies

Small magnets (I used (4), roughly 5/32" [sold as 4mm] diameter, cheap online straight from China)
Thin (I used 3/8" or 10mm) wood (real, MDF, whatever you have access to)
Thinner wood (I used 4mm - about 5/32" - plywood, could probably use good cardboard)
Dowel rod (I used 1/4" (6mm) diameter, 10" (250mm) long
Tiny hinges (mine were super-cheap on the usual online auction site, I want to say $1-2USD for a few dozen "brass" [brass-colour, cheap steel] hinges from China at about 5/32x5/16" or 4x8mm each, but use whatever you can source cheaply).
Some strong glue (super glue, epoxy, whatever floats your boat)
Some wood glue (or cheap school glue)

Optional Supplies:

Felt (to make it quieter)
Leather (instead of or in addition to the felt, to make it even quieter - whatever you have easy access to)
Short LED light strips (single colour, RGB, whatever you have, I just used because I didn't do it for aesthetics, I did it because I play in a darkened room & my eyes aren't what they used to be)
Power source for above LEDs (I used a standard 'wall wart' of the voltage the LEDs were rated for, to keep it easy and not have to buy replacement batteries)

Step 1: Why This Dice Tower Among the Dozens (hundreds?) on Instructables?

Well, I think it has a couple nice features.

I like the action so you can load it up in advance & then trigger the trapdoor when you're ready. I like the quite spacious drawer underneath (probably holds close to 100 dice, I don't have that many to try, maybe more).

It's not in the TinkerCAD (yet, maybe I'll add it), but mine has LEDs to light up the dice because I typically play late at night (nothing fancy, just some short LED light strips). Also not in the Tinkercad, but I lined mine with felt on all the vertical pieces and felt + leather on all the horizontal pieces (so I don't wake people playing late).

Anyway, I made a couple animated GIFs to hopefully inspire someone (I look at Instructables generally less for a complete idea and more for a piece to use in my existing idea).

Step 2: Basic Thoughts in Lieu of Steps

Don't try to duplicate my tower (I mean, unless you really want to) - instead pick up an idea from mine (or rule out an idea based on mine, haha).

I thought about using an Arduino and servo for the action, but in the end simple dowel rod with magnets was way easier.

My local dollar store sells felt, but if yours doesn't, your local sewing shop may have some felt or thick fabric (like a fleece or something) in the discount bin (end of a bolt, etc.), so check that before spending a fortune online or at the craft store. If you can get felt with adhesive already applied (like a large sheet of felt tape) for close enough to the same price, consider getting that, it's easier than felt + glue to get great results.

I used leather because I got it super-cheap as scraps (I think I used to pay $2USD/lb, I have bags of the stuff now). I glued felt to the angled pieces and then leather over that to deaden the sound as much as I could. Not sure how much difference it made and I wouldn't have spent much on the leather to do it.

My LEDs were nothing fancy - I have RGB ones on hand, but I wasn't looking for an aesthetic, I was looking for putting more light on the dice. A thought about that: unless your dice are more transparent than mine, if you do LEDs, you'll want to mount them as high as you can in the area the dice land so that they shine on the top (otherwise, you're adding light without adding contrast for readability).

My hinges came with nails (listing said screws, heh), but when you're using super-thin wood, nails and screws aren't ideal. One option is to add a second layer of wood just where the nails or screws are (on top), to give them something to 'bite' but I think a couple drops of superglue (epoxy, whatever) is easier. You can still attach them to the thicker wood with nails/screws if you like, but if you use a fake wood product (MDF, particle board) you might be better off with glue. If you use the nails/screws, consider using glue too.

To cut the hole for the dowel, keep in mind the most important thing is that the dowel can swing freely. If you want to do it like the Tinker, consider attaching that side after the trap doors and hold/tape a small pencil where you'd put the dowel, swing the door & you have a center line where you need to cut.

I'm happy to answer questions when I can, have fun!

Direct links to animated images:

Drawer
Trap doors

BTW: The different colours are to make it easier to see individual pieces, they don't mean anything more than that.