#7 If you understand the internal schematic of 555 uses an open collector switch for the DISCHARGE output, then you will readily see that is your main difficulty.
As
audioguru said in #7 , you are missing the pullup resistor.
That value should be lower than the Pot value but not so low that the transistor cannot pull down near < 50mV This is generally 5 % of the base resistor when driven from the same voltage as the pullup. It could be 2% to 10% of Rb but that is my guesstimate..
I disagree with the use of the diode, because you want the average voltage of the discharge output to be 50% of Vcc +/- 2% or so then you will always get 50% duty cycle. The diode raises the lower level by 0.6V which offsets the average DC level of the triangle wave which would want to be 50% as the toggle thresholds for set/reset are 1/3 and 2/3 which means the hysteresis is 50% of Vcc.
Coincidentally this the same hysteresis used in CMOS Schmitt Triggers, although the threshold voltages are not controlled by resistor ratios but by Vgs(th) which have a 50% tolerance.
Here you can see your schematic shown in a more logical arrangement
with the pullup and
without the so-called 50% improvement tweak diode, which I think is no help
Notice I am using 15V so the thresholds are nice round numbers 5V ,10V and the average is 7.5V. Then examine the plots and look for the average Vdc and the duty cycle error.