While I am not sure of what "kinetic energy separate from velocity" means$^*$ (I will wait for clarification), there seems to be some confusion as to why wind makes one feel colder, even though the air is moving faster relative to your skin than still air would be.
First, if we are stationary in a location with no wind and air that is at a temperature that is lower than our body temperature, then we essentially will reach a steady-state situation where the air right next to our body is at the same temperature as our body, and the air will become colder as we look farther from our body. Thus, we will essentially develop a "protective layer" of air that insulates us from the colder air around us.$^{**}$
However, if we start moving, or if the wind starts blowing, then this warm protective layer of air will become thinner/non-existent, and the air closer to your body becomes colder, hence you feel colder as well.
It sounds like your science teacher just doesn't know what is happening in this situation, and so they are just saying things that might sound correct until you appear to be satisfied with the explanation so they don't have to be on the spot anymore.
Aside: Something to keep in mind is that humans are horrible thermometers. The classic example showing this is how conductors like metals feel in comparison to insulators like fabric. If we have a conductor and an insulator both at the same temperature, say lower than body temperature, we will feel the conductor to be colder than the fabric. This is not because they are at different temperatures, but it is because the conductor absorbs energy at a faster rate, and so we feel it as "colder".
$^*$ Kinetic energy is defined as $\frac12mv^2$, so I cannot see how kinetic energy can be separate from velocity.
$^{**}$ It is not a perfect insulation, of course, but it is better than nothing.