alt Hacker NewsI should have said that they don't exist in the written language; but using an alternative one lets them be written. This is why romaji appears handy in this matter. It lets us have a stem ending in "r", which doesn't change when we have "a", "u", etc after it.
On the matter of 'si' and 'ti', they are not appropriate in a model of modern Japanese, which recognizes e.g. the onset of ち as a phoneme distinct from the onset of と. The orthography uses a digraph / diacritic for this, writing "ちゅ" for the single syllable that belongs in the -u column of the row containing ち, but that's not a reflection of Japanese phonology; it's a historical artifact.
It's still the case that Japanese speakers have difficulty producing the hypothetical sound 'si', but that doesn't mean that the syllable which is notionally given that place in the kana table represents that hypothetical sound. In English we have the very similar rule that the cluster /sj/ may be reduced to /ʃ/, but this obviously doesn't prove that the word "sheep" begins with the phoneme /s/.