I'm not a native English speaker, but sometimes I get the feeling that the pronunciation of English words is random.
Why is "Yosemite" is pronounced as "Yoh-Sem-Ee-Tee" and written as "Yosemite" and not as "Yosemitee"?
From Dan Anderson, "Origin of the Word Yosemite" (posted December 2004; updated July 2011):
Yohhe'meti (Southern Miwok) or Yos.s.e'meti (Central Miwok) originally referred to the Indian tribe that lived in Yosemite Valley. Yosemite means literally “those who kill” (Yos, “to kill,” the modifier e, “one who,” and the plural suffix -meti). It was used by the surrounding Miwok tribes. The Yosemite people were referred to as killers by these surrounding tribes, who feared them. The Yosemite tribe, led by Chief Tenaya, were composed of renegades from multiple tribes, including Mono Paiute from the eastern Sierra. The Paiute were traditional enemies of the more-peaceful Miwok people.
This summary of the word's history before it reached English explains why the Miwok people pronounced the final vowel-ending syllable of the name as in "-tee," but it doesn't explain why English speakers transcribed the name as Yosemite rather than, say, Yosimmity or Yosimiti or Yosimmatee. Madison Beeler, “Yosemite and Tamalpais,” Names [Journal of the American Name Society] (September 1955)—quoted on the same website as above—suggests that the spelling may reflect the influence of Spanish speakers, for whom the final e would be a voiced vowel roughly equivalent to the Miwok final i sound.
Actually, Beeler doesn't directly address the question of why Yosemite ends in an e, but he does cite Spanish speakers as perhaps influencing the choice of e as the second-syllable vowel in the word:
The name of Yosemite has been connected with the Sierra Miwok word for ‘(grizzly-)bear’ and with a collective noun meaning ‘the killers’ or ‘a band of killers.’ In Mrs. [Lucy Shepard] Freeland’s “Language of the Sierra Miwok” ‘bear’ appears as ïšï'·mati (p. 3) and yošé-·met^i is defined as ‘the Killers’ (p. 159). These words, although of course distinct in the native tongue, show a notable degree of similarity in their phonological structure, and a confusion between them, or their being taken as variant forms of a single term is easily understandable on the part of those unfamiliar with the language. That the second, and not the first, is the true etymon is most likely. In the first place, the word for ‘bear’ lacks the initial y-, and the high central vowel (transcribed ï) of its initial syllable was commonly equated by Spanish speakers with their vowel e and by Anglo-Americans either with ü or with u. And secondly, yošé-met^i fits readily into the morphological pattern of Miwok: the suffixation to the verbal root yóš- ‘to kill’ of the morpheme -e- ‘one who, that which, does something’ yields the agent noun yošé· ‘killer,’ and this is pluralized by the addition of -met^i, “a pluralizing suffix, probably more or less collective in meaning, used in all dialects” (op. cit., p. 158). The variant yohé·met^i?, proper to the southern or Mariposa dialect of Sierra Miwok, in whose range the famous valley lies, indicates that the word first was heard by white men from the lips of speakers of the more northerly dialects. “This [the form with -š-] is the name which was applied by the neighboring Miwok to the dwellers in Yosemite Valley, who appear to have been (at least in historic times) a band of renegades from various tribes” (op cit. p. 159).
Spanish language influence is relevant, of course, because the earliest European colonists in California were from Spain or from Spanish-speaking Mexico.