When I'm rolling, my road bike is definitely faster than my hybrid by about 2-3 mph on average. However once I factor in traffic and other things that slow me down, the difference on a 10 mile commute is somewhere in the 3 minute range. Measurable, but not really significant enough to allow me to sleep later in the morning.
I didn't really notice much of a speed difference when I went to clipless. I like riding clipless, and I think there are benefits, but I don't think you'd notice a huge speed increase.
Possible option Cs: Better fitness, e-bike or stop worrying about it.