No, the original Pi B can play 1080p video just fine. The video is not decoded by the CPU. H264 and older codecs play just fine. It cannot handle h265 videos as it doesn’t have a hardware decoder for those. Kodi works just fine in fact. The interface is a little bit slow, but actual video playback is fine.
I still rock XMMS almost every day. It simply does the job I want it to do and it’s so lightweight it might run on a toaster.
If you want something more modern, qmmp should be great.