It sounds like you are asking the cylinder 'stat to do something it isn't meant to do.
The purpose of a cylinder 'stat is to identify when the hot cylinder cannot absorb more of the boiler's output, not to control the water temperature at the outlet.
Cylinder 'stats are not exactly high precision devices in the first place.
They are made with a fairly wide hysteresis loop to stop them switching the boiler on and off every time you glance at them, and in their usual position, towards the bottom of the cylinder, they only reflect the temperature of that stratum of water in the cylinder - the water near the outlet may be much hotter, or it may be nearly the same as the lower strata. That depends on other factors, including how well insulated the cylinder and pipework is, how long the system is allowed to stabilise after hot water is drawn off, the amount of water drawn off etc. etc.
The hysteresis loop of a 'stat often seems to grow wider, and the accuracy of calibration drifts as they age, so a 'like for like' replacement may help, but do bear in mind that controlling the temperature at the point of delivery is not the primary purpose of the 'stat.
If you really need that level of control, then, as ratonnastick said, a TMV or similar would be the answer.