Guest,
From Ottawa, ON in mid 1960s, our version of Mary Mack has one more line:
After:
To see the elephants elephants elephants
Goes:
Inside the tent tents tents.

We used the same musical phrase for the first four lines, which I would notate as 5 6 7 8_ 8_ 8 (not 1 2 3 4 4 4) because of how it fits with the B musical phrase, two lines long and used for the next four lines:
3 2 1 2_ 2_ 2
4 3 2 1_ 1_ 1

Then back to A tune, down to 5 to start the 9th line. But sometimes we stayed on the pitch of 1 and made a key change by using it as the 5 in the A tune, till after a while we were squeaking ridiculously high and ending in gales of laughter.

I never heard boys sing this. Miss Mary Mack was definitely a girls clapping song in my neighbourhood.

Most of our playground games escaped the safety police, but one serious contact game, Red Rover, was banned for the rest of the school year after someone broke an arm. The following year we played it again after promising we'd no longer launch and and hurl ourselves at the arms through which we intended to break, but keep feet on the ground at all times. Yes, sometimes we forgot those earnest intentions. Red Rover was definitely never played at a birthday party though. Parents in control of parties never scheduled it and we never asked, probably knowing the physicality, especially among girls, would spook them.

Party games were much more civilized: Pin the Tail on the Donkey, Guess When a Minute Has Passed, 3-legged races, egg-races. (Often the first activity was to fold pieces of newsprint into party hats and paint them.)

In my area I don't think anyone older than kindergarten would've been caught playing ring around the rosy unless amusing a younger child.