According to https://arcadeblogger.com/2021/01/31/anatomy-of-arcade-high-score-tables/ who is quoting Atari’s Steve Calfee, the **main** reason for the 3-character-limit was to prevent profanity.
English has a lot of colorful 4-letter-words.
Memory is a factor, but it is an arbitrary factor. There is no hardware reason why allowing 3 would save you an incredible amount of memory over allowing 4. *Usually* a high score entry (that is Initials + Score) will keep it under 12 bytes; but that kind of limitation would also quickly have become legacy. If an arcade machine maker would have wanted to allow 4 characters, they would have been able to. The ‘profanity prevention’ explanation makes a whole lot more sense.
There’s also the issue of legibility and competetiveness. Fewer characters means more easily readable high score tables with more space for entries and more space for larger score values, making players more competetive, making more money.
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