1/ Snobism, as, like you said, you love to walk in the street and no one has the same watch than you...
It is not only about independants, but it is easier with Independants to not have everybody's watch, as the production is, by definition, very limited.
That is a dangerous criteria, ad it may also be related to the feeling ( or the willing ) to belong to an Elite, but well, I prefer not to develop...
2/ Fresh Air, because some of them offer us some original watches you will probably never see with some big brands.
Here, I think about MBF, for example, or Urwerk?
3/ You want something which is closer to your tastes in horology, which can't be made by a big brand.
An exceptional finish on the movement, a personalized dial ... ( K.V is an excellent example ).
Best,
Nicolas