Why doesn't a halide anion react with primary or secondary alcohols using SN2?
The issue is that when carrying out the Lucas test you have a polar protic solution (due to the alcohol which you are testing), in which case Cl$^-$ is well solvated and thus not a very good nucleophile. As you are not using a strong nucleophile, you disfavor the S$_\text{N}$2 mechanism, but since you have a primary alcohol, you are not really favoring S$_\text{N}$1 either. Hence, why no reaction happens and the Lucas test works to identify primary alcohols. I think your question of Why is it biased towards an S$_\text{N}$1 reaction? might be a bit misleading because the reaction is neither biased towards S$_\text{N}$1 nor S$_\text{N}$2 for primary alcohols - it simply does not happen at all (at room temperature). In the case of secondary and tertiary alcohols, however, I would definitely say the reaction is biased towards S$_\text{N}$1 since you have polar protic conditions, a weak nucleophile, and a more stable carbocation intermediate. 

Upon raising the temperature, I suspect that the reaction would then proceed via an S$_\text{N}$2 mechanism for a primary alcohol (as, despite not having a strong nucleophile, the barrier would be lower than via S$_\text{N}$1), but remain S$_\text{N}$1 for secondary and tertiary alcohols. Hope this helps!