Short answer: romanization is an arbitrary way to map Chinese sounds to a roman alphabet characters, so they decided to use X for SH. For convenience, efforts were made to keep as many sounds the same or close to the English sounds those letters represent, but some letters were repurposed.
There have been other romanizations in the past, for example “Wade-Giles” which renders 北京 as “Peking” instead of “Beijing” and “Wade-Giles” as “Wei Chai”. It’s not so much that Messrs Wade and Giles didn’t hear well (although there is some of that) but they used a dialect of Mandarin that had a “k” sound instead of the “j” of the modern standard Mandarin. Also that they were perhaps favoring sounds that English speakers could more easily make. It’s no longer used much (if at all) except in historic things like older restaurant names in the U.S.
In Japanese romanization, there are similar oddities: “si” is used in some texts to represent the し/シ sound, but it’s more commonly written as “shi” these days, which is how it sounds to an English speaker. Any word that’s romanized with an “r” could be pronounced almost as a “l”, because the actual sound spoken in Japanese is in between: ra, ri, ru, re, ro can sound like la, li, lu, le, lo. English speakers have trouble hearing and saying this sound correctly, and Japanese speakers have trouble hearing and saying the difference between “glass” and “grass” because in Japanese there is no difference between those consonants.
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