Classic Arts apparently isn't able to post any broadcasts to their website, so they aren't dealing with Internet rights issues and royalties.
CAS needs to deal with rights and royalties just like anyone using someone else's intellectual property. They have negotiated agreements with the video companies whose work they broadcast (you see the credits at the beginning and end of each clip) -- the companies consider this to be a form of promotion or advertising, so are likely to be very generous with their material, but it is still under copyright and needs to be treated that way.
As far as live broadcast or complete works are concerned, I'm not sure it's a question of ability -- they were designed to use existing commercial materials, and do it in an excerpted, rotating manner. They describe these broadcasts as promotion for the arts in general and live events in particular, and purposefully mix genres all the time. It's a great service, but it's not a substitute for the work that PBS has done in the past to record dancework being made today, and distribute it widely. It's probably better to compare PBS to Kultur and other companies that create videoprograms -- PBS creates the programs and distributes them as well.