In order to solve the current problems of time-consuming manual work and low harvesting efficiency during the harvesting of potato breeding trials and the inability of traditional harvesting machinery to meet the requirements of potato single plant harvesting, a single plant harvester for potato breeding with a combined excavation device and a conveying and separating device with alternating concave and straight units is designed. This paper analyzes the force, motion, and collision of potato plants in the digging transition stage and conveying and separating stage. The key factors affecting the rate of single plant integrity and the rate of injured potatoes are determined. A three-factor quadratic regression orthogonal rotating combination experiment is conducted with the rate of single plant integrity and the rate of injured potatoes as experiment indexes, forward speed, conveying speed to forward speed ratio, and the rotational speed of the rear-vibration device as influencing factors. Based on the experiment results, a mathematical regression model is established, and response surface analysis and parametric analysis are conducted to determine that when the forward speed is 0.65 m∙s−1, the ratio of conveying speed to forward speed is 0.98, and the speed of the post-vibration device is 11.52 rad∙s−1, the single plant integrity rate is 97.14% and the rate of injured potatoes is 0.67%. The machine can better meet the requirements of potato single plant harvesting and solve the problem of plant mixing. From the perspective of potato damage rate, potato breeding harvesters are also suitable for traditional potato harvesting.