... why does "beady-eyed" have such negative connotations in English?
I don't think it does, particularly. There may be contexts in which
someone might be referred to as "beady-eyed" is they were thought to be
scrutinizing something more closely than they should ... but I would
not have thought that such a context was typical for the phrase.
In general I think that to call someone beady-eyed is merely to suggest
that they have keen eyesight or good powers of observation, which is
surely a positive rather than a negative comment.
If I had written something and wanted a friend or colleague to
proofread it for errors I might ask "would you mind casting your beady
eyes over that?" with the implied compliment that I thought the person
in question would quickly spot any error I might have made.
Cheers,
Daniel.