Somebody sent me these links. The first is the company
PermissionTV, what I would describe as a for-profit and higher quality YouTube. They offer their services to libraries, an
example setup here and
Orange County's service through them here. The presentation is miles above anything an individual library would do but it could also be miles above what a lot of libraries would need to do.
Speaking from the POV of a middle-sized library you're going to be able to do what you want through YouTube, especially with the ability to insert the video into your website with just a little code. The only real question is how long YouTube can remain free and if there's an eventual cost what will it be?
Larger, better financed libraries, or library systems, will probably be more interested in what P-TV offers.