You can't simply take a characater apart and glue meanings of the parts together - it's more complicated. Having "乞" part doesn't necessarily mean it has the meaning of begging - it's a "sound", rather than meaning, element, which "provides" the sound of "吃".
And although I don't know the "喰" character, I can tell it didn't become Chinese 吃 and Japanese 食. 食 is more "ancient", where "吃" seems only used so widely in modern time.
Simplification of Chinese characters indeed started many arguments, but the "tranditional" Chinese used in Taiwan has also developed some "simplified" characters.
Yeah the Japanese version of hànzì is much more ancient (Song or Tang dynasties, I think) and have changed comparatively little since then. Also, they simplified some characters in a very different way than how the Chinese did.
(I'm probably wrong but: if I remember correctly, Japanese has changed a lot, but the writing system hasn't so the sounds like part of the character isn't always correct or even close. It's a lot like Irish or English in that the language has changed much but the writing hasn't.)
And although I don't know the "喰" character, I can tell it didn't become Chinese 吃 and Japanese 食. 食 is more "ancient", where "吃" seems only used so widely in modern time.
Simplification of Chinese characters indeed started many arguments, but the "tranditional" Chinese used in Taiwan has also developed some "simplified" characters.