I’m fairly sure that “wallball” played in most elementary schools around the US is a different game than what is being referred to here. Growing up in New England, what we called wallball involved throwing a tennis ball against a wall and attempting to catch it before someone else, while the NYC handball/wallball is closer to racquetball, but played outside against a (usually) freestanding concrete wall with palms used instead of racquets.
We played both handball and 4 square in southern california, though I don't recall it being a thing at the school I attended in the midwest, nor do I remember having played it in virginia.
Interesting! The mention of California made me think that was likely but Florida had the opposite effect. In terms of culture trends, Utah seems to lag behind California by 2-3 years (from my lay-observations) but I’m not sure about Florida. The geographical distance had me thinking it was a more-likely national phenomenon.
To continue with Kid's games, we used to play 'JailBreak' a kind of tag/capture the flag. you'd try to get your fellows, who were trapped out of jail, if you sneaked up and tagged them with your hand without getting tagged and captured. We played it in the evening sometimes. That was in suburban NJ near Philadelphia.
Yeah, that reminds me of part of how I played capture the flag as a kid, rather than its own game. One could be “caught” (tagged) and jailed if they were on the opposing side looking for the flag and they had to be tagged by a teammate while in jail to escape (still free game until you’re on the friendly side of the line).