MPEG-TS is incorporated in many digital TV standards. They will continue to be around for a long time simply because of that, regardless of technical points.
Right, but transport stream should only be used in broadcasting. It shouldn't be used on discs (ahem Blu-ray) or other storage and it shouldn't be used over the Internet. Program stream is usually a better choice.
I thought Blu-Ray switched to program stream, but DVD was transport stream (sounds like I've got things backwards though). Transport stream for optical media isn't that insane though, because while you can seek, seeking isn't great, and if you have a failed read, it's probably better to just move on than to retry that sector a bunch of times. Of course, using transport streams allows for smaller buffers and lower BOM, so there's that too.