The carburetors on small engines do not have anything like the accelerator pump found on carbureted auto engines. Theh purpose of that pump was to increase the mixture richness when the throttle is opened suddenly.
When you engage the PTO, this places a load on the engine that the governor instantly senses and abruptly opens the throttle to produce the additional power needed to keep the engine from losing RPM. But there is no accelerator pump to enrich the mixture to keep the engine from stumbling and the additional air admitted by the throttle opening causes a lean condition. The is most apparent when the engine is not thoroughly warmed up.
The Cub Cadet shown in my avatar has had this problem since the day it was new. My 'solution' to this problem is to pull the choke out at the same time I engage the PTO, then push it back in when the blades can be heard spinning. Once it's warmed up, I don't have to do this.