I believe the slots for the longitudinal pieces should be cut on the table saw when the glue-up is square, however i would be concerned about tearout of these slots when turning on the latheLooks like the struts lengthwise were inset into small dadoes cut in the blank. May be easiest to cut those in the blank prior to turning the core piece. Alignment of the spindle axis with the dado cuts will be very important.
It's always better to make things yourself rather than to buy ready-made. Also, with a lathe nearby and growing hobby, it presents a challenge that most of us are up for !!Interesting. However, IMHO, to make pasta, ravioli, or something similar, it's easier to buy it ready made. My BIL bought a pasta maker and used it a few times and put it in the cabinet. I do want to make pierogie from scratch as I was told my grandmother made them. Pierogie (pee-ROW-hee) is an item similar to ravioli. They are made in Eastern European countries.