Respect Sir, I share your appreciation of those curves. When I see wood that has been worked in the days before electricity it gives me extra appreciation for their patience.
Discussion
Hand-carved joinery blows my mind. Mortise and tenons, dovetails, etc. So much precision and time required!
I think dowels hit the sweet spot. Strong enough (stronger and more durable than screws) and easy enough to drill in if you have a good enough guide or jig. FEELS like old-school joinery but w/modern conveniences.
For this project I took leftover 2"x2" and 1"x3" pieces and drilled matching guide holes on the drill press. Then clamped to the matching piece and drilled through the guide holes. Worked well enough (the table actually stands flush on all four legs!!! ). The guide holes did get looser with each pass with the drill, but lasted long enough to finish the job.
Now that I'm convinced that dowel joinery is the way to go, I invested in a somewhat expensive but awesome dowel jig to make everything easier and faster next time.