What we know about audio and video clips of shaykhs on YouTube and other sites is that they are not usually subject to copyright, and that they are free for whoever wants to download them and put them on various websites. In fact most of them are originally taken from free satellite channels or Friday khutbahs or lessons that are free and open to all. In this regard there is nothing wrong with publishing those clips on other websites in order to spread the benefits thereof, if that is the case.
But if they are subject to copyright, then it is not permissible to publish them except with the permission of the authors. Most websites allow publishing their material subject to two conditions:
1. that it is not used for commercial purposes
2. that it is not changed or altered.
As you do not want to use the material you take for commercial purposes and you have not changed the contents of this material, there is nothing wrong with republishing it and spreading the benefits thereof, in sha Allah.
As that is the case, there is also nothing wrong with putting the name of your website on this material so that people will know about it and so as to promote the beneficial material on the website.
If you can put on your website a section for the mukhtaaraat that you have taken from others, and have not produced yourself, that is better and is further removed from claiming as your own that which you did not produce itself. In this way you will be able to rid yourself of any discomfort you are feeling whilst also promoting your website.
And Allah knows best.