Are cheap Amazon standing desks actually stable?

It depends on the frame design. Across specialist standing-desk subreddits, the recurring theme is that single-stage and two-stage budget frames are stable when sitting but can wobble at full standing height, especially with monitor arms or heavier loads. Reviewers repeatedly suggest stepping up to a three-stage frame if rock-solid stability at standing height matters to you. Two-piece (splice-board) tops are also flagged as a stability weak point.