MacBook Air gets very hot above the keyboard and near the logo when watching YouTube or gaming

Make an appointment at your local Apple Authorized Service Provider and have it professionally diagnosed.

It is not unexpected that the area immediately surrounding the processor of any computer will become very warm or even hot when used under load. Unlike the MacBook Pro or any other Mac, the MacBook Air is a fan-less design that relies on convection and radiant cooling and uses the case and chassis as part of its cooling system. Gaming and video playback are two uses that can cause the temperature to increase.

User @rkaufmann87 has provided great guidance when they suggest having the computer diagnosed by Apple or an Apple Authorized Service Provider. You can make a Genius Bar appointment for hardware/software service and support using this link: Contact Apple Support - Apple Support

Meanwhile, please take a look at this guidance from Apple about how to keep your cool:

Keep your Mac laptop within acceptable operating temperatures - Apple Support

Some of the tips provided in that link include updating the OS to the latest version available and using Activity Monitor app to look for "malfunctioning processes".