At a previous church where I served we used to put a list of YouTube links on the church's web site each week.
Alex, we've implemented something sort of like what you talked about - not on a weekly basis, but we have a oage on our church website where we have our current band repertoire - 25 to 30 songs. So thank you for the idea!
For the songs where there are youtube videos, we post a link to the video, which I understand is completely legal, you can post a link to anything that is on youtube. The artist may ask youtube to take down the video itself, but that seems to be pretty rare, most people doing worship music seem happy to have links up. Since our band doesn't try to do anything "like the CD," I'm not concerned about finding the original video, just whatever one inspired us to learn the song. Also, I make a point of looking for the shortest version of the song I can find - the three minute "studio version" rather than a nine-minute live one.
For my own original songs, I put up an mp3 on my own webspace. For any "unpublished" songs (like things I get from WtR), if there is a soundcloud version, I'll point to that, or if the songwriter is okay with it, I'll record the audio from their version and put that on my own webspace.
All of this was done kinda in response to some suggestions that we should display notated music for our songs, which creates a lot of extra work and permission-gathering (and did not seem to help our congregation learn the songs, although I'll admit, I didn't want to do it from the start anyway). Also, the last time we taught a new song to the congregation, we just put a printed version of the lyrics in the bulletin that week and that way the songleader could say, "let's sing that chorus again..." without the person running MS having to jump around and predict what we're going to do next.
...is the page on our church website, it's kind of out of date right now, some of these songs would be replaced by others, etc. I send the changes to the church office and they make the updates, usually done within a day. I have good intentions of updating all this stuff once I've planned out the music for the new year.