You asked "Do kids these days still sing these rhymes? Or is it all computers & video games now?"
While I'm neither a playground supervisor or [any longer] a parent of young kids, I did stay at a Holiday Inn last night {sorry, that's an American joke which others may not get]...
For a number of years I have been collecting children's rhymes and I can state without reservation that children still do counting out rhymes, handclapping rhymes and other types of rhymes.
Besides a number of Mudcat threads, there are quite a few othre websites where people can post examples of rhymes. On some of these websites more than others, it appears that children and youth-as well as adults-post ecamples of rhymes. Imo, the Internet is helping to perserve examples of rhymes, and also is make individuals aware that there are versions of specific rhymes other than the one that a they know. In that regard, I believe that the Internet may influence changes in rhymes-as children may drop their version in favor of vrevise their version or add trhymeis helping to infulence spread er another one or add to their version of their rhyme. That's the folk process in action.
Internet threads such as this also provide an opportunity for folks to consider the origin, source materials, and meanings of specif children's rhymes-and the way that their performance activities may have changed over time or in different geographical areas, and/or among different populations.
WIth regard to the "beestay" example that you cited, I wonder if this rhyme has a Spanish language source and are the changes in some if its words as result of folk etymology?
For instance, was "beestay" originally "como la vista"?