There is a mixture of right and wrong here. HTML5 does not have the typical plugins, it does have API layers. Videos in HTML5 don't actually use HTML5 code, they use CSS3 and Javascript for video playback, which then in turn use codecs. Depending on your system, the GPU may have built in codecs that allow for accelerated decoding of the content, but not all graphics processors have acceleration for all codecs. In fact very few have acceleration for VP9 used by Youtube by default now, where as your GPU should have acceleration for H.264 which is likely what Flash Player defaulted too. For VP9, your CPU would be stuck doing the decoding, and VP9 is more intensive on CPU/GPU to begin since they are making more work for the CPU/GPU putting the image back together in order to have greater bandwidth. So you are in one of the cases where you want Flash, or at least to set HTML5 to use H.264, and not to use VP9 which most of the world could do.
You should be able to change this setting on Youtube somewhere.