I turn vases and candle holders from whole logs or quarter-split logs, substantially longer than the final product, and not necessarily dry. Square stock would work almost the same.
I mount the workpiece between centers, and turn a tenon to match the chuck jaws. Then I mount it in the chuck, and bore the hole from the tailstock. For candle holders, I use a spade bit ground to the shape of a candle cup (e.g. CSUSA's 041-0444).
I replace the drill chuck with the live center, with the truncated cone/cup center just nesting in the hole, with very little end pressure. Turn the final shape. I shape most of the bottom by cutting into the excess length to leave a foot ring, until there's a small post remaining. Cut it by hand with a small saw, and remove the remnant with a hand chisel or a small router bit in a Dremel, off the lathe.
If the tenon is the only intended waste, I'd cut it off flush. For hand saw clearance and rotational symmetry, I'd re-chuck the tenon with reduced purchase, and rotate the piece by hand, sawing gradually to the center.
A foot ring is more stable than flat, but with dry wood it shouldn't matter.