Video is not usually DXT-compressed, but using other methods (which not only compress a single frame but also use deltas between frames). It is of course possible to use DXT to compress individual video frames, and since that is a simple solution it might just work for some cases.
I can see how DXT encoding is nice for textures you want to use for a longer time, but is it really needed to do all that encoding on CPU for a video when you display each frame only once and then throw it away?
As to why one would want to do such a silly amount of work for a frame that one only watches for a split second, the answer is that the amount of data you need to stream in for video is forbidding if you possibly ever want to do anything else. It may be overall forbidding too, depensing on resolution and frame rate.
Let's say you want to play a 480x270 video (1/16 the size of 1080p) at 25fps. Assuming 32bpp, that's 138MiB/s of data. Try and find a harddisk that consistenly delivers at such a rate (my SSD has no trouble doing that, but sadly not every consumer has a SSD just yet). And now imagine you maybe want a somewhat larger video. Make it only twice as large in every dimension, and you reach the theoretical maximum of SATA-600.
Compression will reduce this immense bandwidth pressure by a factor of maybe 500 or 1000, which is just what you need.
About doing YUV->RGB on the GPU, there is no objection to that.