I've always liked square platters. These are particularly nice. I like how you cut the blanks to size and shape first... I found out within my first two square platters, that if you don't cut them to the proper size and shape first, you'll usually end up with an out of square shape, as blanks are often not truly square (even if they were originally cut that way, if it was green wood, then it will move as it dries.)
I have a bandsaw. Don't have any of the sanding tools. You think cutting a blank to size and shape like that could be done without the drum or disc sander?
Thanks! That's one my most popular designs. The tops of these are gently dished. People use them for cookies, cheese, coins, wallets, etc. Make great wedding presents, house warming gifts. I have someone coming this weekend to get one.
I draw the arcs for the rim carefully and if I don't cut very close to the line, it is difficult to the the edge of the rim even all the way around, a lot easier if the starting shape is right. (I only turn dry wood so nothing moves significantly.) I've made so many of these I made a set of carefully shaped plexiglas templates of various sizes (from 7" to 10") and trace them onto the wood before bandsawing. (The clear plastic is nice since I can move the template around on the wood and avoid defects, get the figure the way I want, etc.)

When I do demos on these I always take some templates and poster board so those interested can trace the outline to take home and not have to recreate the pattern.
I think, yes, if you cut fairly close to the lines with the bandsaw you can turn everything then smooth the edges with sandpaper, soft sanding blocks, and or sanding sticks.
As well as gently rounding the edges, I often round off the point on the corners a little.
However, even a cheap stationary belt/disk sander can be a great help for a woodturner. (I use the disks more than the belts) I have them in sizes from 1" belt/6" disk to 6" belt/10" disk. Another very easy thing people do is make their own disk sander from a piece of plywood held in a chuck or better, mounted on a faceplate on the lathe. Some build a plywood table made 90-deg to the disk. I mostly use coarse grit, 80 or 60.
JKJ