Well, Legolas says they 'do not/need not count the running years, not for themselves, so the issue is not whether they can or can't count the running years per se - obviously they can do that - but whether they count them for themselves. To say they do not count them for themselves implies either something in their nature - the 'running years' do not register on them personally - or, that they have made a deliberate descision not to count them for themselves.
Changing it to need not means either that the 'running years' do register on them but that they can somehow ignore that, or that its all dependent on circumstances - sometimes they'll register the passing years, sometimes they won't.
It changes a 'definite' into an 'indefinite', a 'certainly' into a 'maybe', a 'will' into a 'perhaps', & so it alters completely what Legolas is saying.
My own suspiscion is that when Tolkien came to read over what Christopher had written down (with the missing word) he wrote in what seemed the obviously 'correct' word, whatever his original idea had been.
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