If you have good internet, you don't have to download the Videos, but it would make your setup more mobile and fool-proof. I don't see an easy way to do pitch control on videos. (others may have an idea...I don't)
You can connect the output of your computer/smart TV to a mixing board along with one/more microphones. The output of the mixer goes to your PA ( amp/speaker or powered speaker). If you don't have a mixer, Amazon has relatively inexpensive Behringer and Yahama units with effects that I've used and recommend.
https://www.amazon.com/Behringer-XENYX-QX602MP3-6-Channel-Player/dp/B01B7QKLWC
You don't need karaoke software as you aren't using karaoke files. Some karaoke software does have the means of playing videos but that just complicates things. If you are playing the files online, you can just use your browser. If you want to download them, use a shareware video downloader like 4k Video downloader ( https://www.4kdownload.com/products/product-videodownloader ) and a video player like VLC. I've used the free version of 4k Video downloader and it works fine without having to purchase the "pro" version. Let me know if that's not clear and I'll try to help.