The article local model structure on simplicial presheaves states that for a site with enough points stalkwise weak equivalences of simplicial presheaves coincide with weak equivalences of simplicial presheaves in the Bousfield localization of componentwise weak equivalences with respect to all hypercovers (i.e., weak equivalences in a local model structure).

Is a proof of this statement written up somewhere? (The article cited above gives a reference to Jardine, which claims, but does not prove this statement.)

Also, is it possible to formulate an analog of this statement for sites that do not have enough points? (Presumably we would have to talk about sufficiently refined (hyper)covers instead of points.)

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