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In Modern English, the class of nouns that start with /v/ and /z/ has been expanded due to loanwords from sources like French and Greek. But none of the main source languages for English loanwords have [ð]. (Modern Greek does, and Spanish has [ð] as a non-phonemic allophone of /d/ (though not word-initially), but I don't know of any loanwords to English where this is reflected).

In Modern English, the classes of nouns that start with /v/ and /z/ have been expanded due to loanwords from sources like French and Greek. But none of the main source languages for English loanwords have /ð/. (Modern Greek does, and Spanish has [ð] as a non-phonemic allophone of /d/ (though not utterance-initially), but I don't know of any loanwords to English where this is reflected).

There might be exceptions to the statement in the title of your question, but I'm not going to quibble. The simplest reason for the lack or scarcity of word-initial ð in English (for all word classes outside of a small closed set of "function words") is because there is no regular historical source for it. In Old English, the unvoiced sounds [s f θ] are believed to have been in complementary distribution with their voiced counterparts [z v ð]; the voiced consonants only occurred when they were both preceded and followed by other voiced sounds (such as between vowels, or after a voiced consonant and before a vowel) and the voiceless consonants occurred elsewhere (before and after voiceless consonants, at the end of words, and also at the start of words). The distribution in Old English may have been different depending on the dialect, and some dialects of Middle English are known to have had voiced initial [v] and [z] at least instead of [f] and [s]. However, while a handful of words with initial /v/ in Modern English come from this source (vat and vixen) I can't find any comparable words with /z/ or /ð/.

There might be exceptions to the statement in the title of your question, but I'm not going to quibble. The simplest reason for the lack or scarcity of word-initial ð in English (for words of all grammatical classes, whether nouns, adjectives or verbs, outside of a small closed set of "function words") is because there is no regular historical source for it. In Old English, the unvoiced sounds [s f θ] are believed to have been in complementary distribution with their voiced counterparts [z v ð]; the voiced consonants only occurred when they were both preceded and followed by other voiced sounds (such as between vowels, or after a voiced consonant and before a vowel) and the voiceless consonants occurred elsewhere (before and after voiceless consonants, at the end of words, and also at the start of words). The distribution in Old English may have been different depending on the dialect, and some dialects of Middle English are known to have had voiced initial [v] and [z] at least instead of [f] and [s]. However, while a handful of words with initial /v/ in Modern English come from this source (vat and vixen) I can't find any comparable words with /z/ or /ð/.

There might be exceptions, but I'm not going to quibble. The simplest reason for the lack or scarcity of word-initial ð in English (for all word classes outside of a small closed set of "function words") is because there is no regular historical source for it. In Old English, the unvoiced sounds [s f θ] are believed to have been in complementary distribution with their voiced counterparts [z v ð]; the voiced consonants only occurred when they were both receded and followed by another voiced sound (such as between vowels, or after a voiced consonant and before a vowel) and the voiceless consonants occurred elsewhere (before and after voiceless consonants, at the end of words, and also at the start of words). The distribution in Old English may have been different depending on the dialect, and some dialects of Middle English are known to have had voiced initial [v] and [z] at least instead of [f] and [s]. However, while a handful of words with initial /v/ in Modern English come from this source (vat and vixen) I can't find any comparable words with /z/ or /ð/.

There might be exceptions to the statement in the title of your question, but I'm not going to quibble. The simplest reason for the lack or scarcity of word-initial ð in English (for all word classes outside of a small closed set of "function words") is because there is no regular historical source for it. In Old English, the unvoiced sounds [s f θ] are believed to have been in complementary distribution with their voiced counterparts [z v ð]; the voiced consonants only occurred when they were both preceded and followed by other voiced sounds (such as between vowels, or after a voiced consonant and before a vowel) and the voiceless consonants occurred elsewhere (before and after voiceless consonants, at the end of words, and also at the start of words). The distribution in Old English may have been different depending on the dialect, and some dialects of Middle English are known to have had voiced initial [v] and [z] at least instead of [f] and [s]. However, while a handful of words with initial /v/ in Modern English come from this source (vat and vixen) I can't find any comparable words with /z/ or /ð/.

If we take a broader look at languages, /ð/ is fairly rare as a distinct phoneme, and it is easily changed into other sounds, but there are definitely languages where it occurs at the start of more words than it does in English (for example, the aforementioned Modern Greek).