Neither Mandarin sh nor x exist in English. X is sharper and is a little more like shy-, whereas sh is darker and a little like shr-. In reality they both strike English speakers as being quite like sh, because we use the space differently.

I've heard various native English speakers who are fluent in Mandarin saying e.g. Xi as Si. I don't know if there's a dialectal difference in English or in Mandarin or both that accounts for it.

> That said, I still wonder why -ui was chosen for "wei"

It looks better, if you have a certain aesthetic.

> "shi" is pronounced "shr"

As far as I know Mandarin speakers just consider them the same. Pinyin is designed for native speakers.

> "-ian" is pronounced "yen"

The e in Pinyin isn't really the English e sound. It's more like the American short u in fun or the NZ short i in pin. It's not obvious what letter to use to represent that, but e is not completely unreasonable. When they do have an e-sound according to the internal logic of Mandarin, it's a variant of an a-sound.

Shi is pronounced Shr? That sounds more like your typical pirate Beijing hua. Which can get quite confusing when you get out of Beijing to let's say to Shanghai business trip and you can't figure if something cost 4 or 10 kuai at street vendor (my personal anecdote, I think they actually do it on purpose since normally 4 would have very different sound).