This use of the apostrophe goes back to Old French, at a point when the final /ə/ of any word was always pronounced, unless it was followed by a vowel, in which case it was systematically elided.
In this respect, this means that le (and any other word used with the apostrophe, like se, que, je, si, de, la or (at the time) ma or sa) would have had the exact same behaviour than une (or any other words that doesn't vary orthographically whether followed by a vowel, a consonant or nothing, like cette, grande, elle or certaine). So any explanation relying on phonology seems suspect or a post facto rationalisation.
There is, however, one factor that unites all the words that show orthographic elision: they're monosyllabic (excepted compounds ending in que, that would have at first been written in a single word). Their lost vowel is their sole vowel, and they end up relying on the following word to exist. By contrast, une is more recognisable as its own unit even when followed by a vowel, since it keeps at least one syllable. This means that there was a reason to write those monosyllabic words as a unit with the word they leant on, with the apostrophe serving as a boundary between the two morphemes, while leaving polysyllabic words separate, even those like elle or une that were also clitic in nature.
In modern French, it makes a bit less sense, since said rule of /ə/ elision before a vowel isn't active anymore, and we drop /ə/ in short monosyllabic morphemes that are dependant on the following word all the time even when said word starts with a vowel. But French orthography isn't made for modern French, but for its long dead progenitor, so such inadequacies are commonplace and inevitable.