If a car headlight's light travels at the speed of light, then does the car technically travel at the speed of light since the headlights are part of a car, and the light is made from the headlights?

If you put it that way then technicaly yes

I'm guessing you have seen the vsause video about this since you are mentioning it but if not you should check it out. The video is a little unclear about this beause scientists are still debating it but I guess yes.