Not that I have anything against desks, but a better support would not have a floppy wood tabletop in the picture. You'd want (1) something higher than a desk, for most 'bench' lathes, you want heavy support down to the (hopefully concrete) floor if you can get it (for vibration damping), and despite the best intentions of furniture makers, the glues of "the 1930's" may be somewhat degraded (and a little shaking could pull the joints apart), Lathe on desktop results in no lathe damage, and bad writing surface on the desk.
Wood pedestals with modern glues, and enough doors that you can hide drawers behind, under an oil-impermeable swarf tray, would be my preference. A rollaround with drawers can be slid between the pedestals.
But, it's your lathe, I always think big when I don't have to build it...