The circuit diagram is made a lot less clear by having
to show four wires joining and having
to show two wires crossing without joining.
This is my slight re-draw that never has four wires joining at a point, and has more visible dots at the joins.
It also has component numbers that are visible and I have called the final output transistor Q3
The circuit gives the same overall behaviour as the relay circuit, but it switches the negative of the centre brake light, not the positive. The centre brake light is connected on the right between LED+ and LED-
When only left is turned on, the current will flow through R1 and R3 to the emitter or Q1. Some of that current will flow through R4 and R5, so Q1 will turn on and the voltage at the emitter of Q1 will remain low. Q2 will be turned on by R6 so Q6 will keep the voltage low at the gate of Q3 and Q3 will be off.
When only right is turned on, Q1 and Q2 remain off. D4 keeps the voltage on it low, so that the base-emitter breakdown voltage of Q1 and Q2 is not exceeded. The emitters of Q1 and Q2 remain at low voltages, and Q3 will be off.
When both left and right are on, D4 and D5 keep the voltages the same on the base and emitter of Q1, so it stays turned off and current can flow through R3 and D5 to turn on Q3 and the brake light comes on.
If left then turns off, with right remaining on, the base voltage of Q2 is kept high from the right signal, via R5 and D7, so C1 remains charged and Q3 stays on.
If right turns off with left remaining on, the base voltage of Q2 is kept high from the left signal, via R1 and D6, so C1 remains charged and Q3 stays on.
When both left and right turn off, the base of Q3 is pulled down by R6 and then C1 is discharged ready for the next operation.
In summary, when both left and right are on, C1 is charged, When both left and right are off, C1 is discharged. If only one of left and right is on, C1 is remains in its previous state. When C1 is charged and either light is on, the brake light will be on.
So I think that the circuit works fine. However, some components need choosing carefully. Q1 and Q2 have their base-emitter junctions reverse-biased by a voltage equal to the zener voltage of D4 and D5. If there is any leakage through Q1 or Q2, that will charge C1. That condition should only go on for a short time, so some leakage won't hurt, but it's not something I am completely happy about.
Also, when left only is on, the circuit relies on Q1 keeping the voltage on its emitter very low, and if the voltage on right doesn't stay very low, then you could get some voltage charging C1