I'm kind of surprised nobody has suggested using hardware acceleration in the browser, like VLC uses.
Back in the olden days of yore when your PC was new and dinosaurs roamed the earth, Youtube used MP4 based on H.263 and even considerably weaker PCs than yours like Pentium 4 or Pentium M could play 1080p videos in software! However as Youtube began to host far more videos they went to more highly compressed H.264 to save on both the storage space and bandwidth required. Obviously this required much more CPU power to decode. Even worse is the even more compressed later H.265 or VP9 (which Youtube chose because they didn't want to pay the licensing fees for H.265)
See, Youtube today is mostly VP9 but has increasingly over the past year been switching over to AV1 (the former VP10 project, mostly on their 4k+ videos). Your Kepler GPU is too old to hardware accelerate either of those, so your poor CPU has to decode them entirely in software
However you are in luck because your GT710 does support H.264 hardware assist, which is what VLC is using. All you have to do is tell Youtube to only send over videos in H.264 format. The most straightforward way to do that is to install the H264ify plugin/extension in Chrome or Firefox, so if the video is available in that format it will be sent to your browser in H.264 to greatly improve performance and reduce the load on your CPU.