Up until just last year "si" "ti' and "tu" were the proper official way to romanize "shi" "chi" and "tsu": https://en.wikipedia.org/wiki/Kunrei-shiki
Are you talking about the v-z-g-l sequence of harsh consonants? That looks uninviting in print, but in practice it's just a quick puff of air.
Erm, wtf? The English "si" sound does exist and sounds different from し. There is a reason people don't want to write Sinzyuku, and while I think they're making the wrong tradeoff, it is a tradeoff and should be acknowledged as such.
Wiktionary lists eight whole words that use it! (The entry for スィ itself, a couple obscure loanwords and proper names, and a couple alternate spellings of words that Japanese people normally pronounce using the "shi" sound.) https://en.wiktionary.org/wiki/Category:Japanese_terms_spell...