I'm confused why this part of circuit is here as its +5 vDC output doesn't connect anywhere else in the overall circuit. Am I missing something or is the circuit incomplete?
The schematic does not show it, but U4 requires and uses the 5 V supplied by the 7805. The schematic is is incomplete, and its author expects its users to know that U4 requires a power connection.
Also, do you people think this is a well designed soft start circuit?
The circuit in the schematic is quite clearly not the exact circuit in the video. An obvious pointer to this fact is that the circuit in the video has a LED, which is not present in the schematic.
Additionally, the circuit in the video does not appear to soft start the lamp, but rather to start it at full intensity and then, after a short bit, to dim it somewhat. However, it is possible this apparent dimming may be an artifact of the video camera adjusting to the sudden brightness.
I would add that the author of the site has a very poor understanding of electronics. To motivate a soft starter, he writes:
This surge current is much higher than the normal load current. Especially for inductive loads. When an inductor is cold or not energized, it works like a short circuit.
That is hogwash. That is not how inductors work. When an inductor is not energized, it impedes (i.e. it resists, in the colloquial sense) a sudden inrush current.
I want to do this as a learning exercise as well as having a practical usage.
Be aware that you would need to write code for, and program U4, the PIC12F675.