I further isolated the problem. I got rid off my external voltage control circuit, now the original voltage control circuit on the power supply's PCB is connected. When the midpoint between R62, R82 and R88 (I consider it midpoint even though there is also S2 switch between R88 and R62,R82) is on the pcb, then there is the ramp up. When I make the midpoint outside of the pcb, then there is no ramp up.
Now it is simple - connect the yellow alligator clip (it connects the midpoint to the PCB) - there is the ramp up. Disconnect the yellow alligator clip - there is no ramp up.
Now, what is the issue with the traces? The midpoint goes also via S2 and is connected to A12 (A3) and A13 (A4) terminals.
EDIT: I noticed I can further isolate the issue by switching S2 to the other position (switching it off). Now when the midpoint created by me outside of the PCB is connected to the side of the S2 switch that goes to R88, CR27, C37, then there is the ramp up. When the midpoint is connected to the other of S2 that goes to R82, R62, C29, there is no ramp up.
EDIT2: Now I have the solution to solve the ramp up issue of the power supply by making the midpoint outside of the pcb (I don't use switches and terminals on the back of the unit). But still I would like to know what causes the issue. However, now I don't know how to find the cause other than cutting physical pcb trace between R88 and S2 in half and isolating the issue further. Visually, the trace looks ok everywhere to me.