I use something called iSkySoft Youtube Downloader. It can cost some money, but only if you want to convert the .flv file. If the software isn't paid, then conversion will give the video a watermark. But if you're happy with the .flv, then it will download and keep the video for you without marking it.
I thought that would be fine. But, here's what I get using iSkySoft:
And they only play using iSkySoft's player. I would prefer a .flv file!
I've already spent hours trying to do what used to be easy and I'm getting too old for this crap (and really fed up with Apple).