This is one instance where it actually isn't entirely a personal preference, manufacturing methods and one's capabilities come into play here as certain things are better suited.
For example, what skills and tools are at your disposal? If you have a CNC handy, well then it makes sense to use that, but if you have no cnc or cad skills, but you have access to a woodshop well wood becomes more attractive. Not every tool works well for everything either, lasers do not like Lexan but love acrylic and wood but while you can laser and cnc wood the way you would create something would be radically different.
Anyhow...
Most people who dislike 3d printing dislike it because of the 3d printing they have seen. If done well and/or worked after it can look every bit as good as injection molding or even cast/milled metal. Better still, in this case since you want low profile you could easily print an undertray and have a metal plate cover the entire top, hiding almost all of the printed parts. This is probably the cheapest, most solid solution. A bit of sanding and primer and that case would look just as good as anything you can buy (sandable primer and scratch filler is your friend) though for sound I think PETG is a better choice but it doesn't hold paint well. For a more solid sound and feel you could also get a metal top and bottom then 3d print or laser cut acrylic for the middle. Personally, I still like the wood idea and it's something almost anyone can do with relatively common hand tools or even a Dremel. All that said, the lowest profile you can get will be the 3d printed base, you can make recesses for the wiring and contacts, something not easily done on the other methods and it isn't conductive even if it contacts, but it won't give you the same sound as solid aluminum, it can be good, but it won't be the same. If you really want that solid sound you need mass, plain and simple, you can deaden it, you can tweak it, you can change it to be more pleasant but nothing absolutely will match mass in every way but you can fake that. For this, the best way to do this is 3d printing, create pockets for coins or lead shot then add a little epoxy to hold it all in place. This probably didn't help you decide and there's more than one way to do this stuff but like I said before, it comes down not just preference but abilities and methods you can take advantage of.
My advice, use whatever method is easiest, fastest and cheapest for you.
Why? Because chances are this will not be the last one you do, in fact it may spur you to immediately try something different once you figure it out and have a chance to try it so doing it cheap, easy and fast means you can redo it better a second time with all the improvments you have in mind. It also means you stop obsessing over the "best way" and get moving on it.