Why is the Lucas test not recommended to differentiate higher alcohols?
The problem is that the alcohols themselves might be insoluble. Ethanol is soluble, yes, but as the alkyl chain length increases, the hydrophobic nature due to the chain starts dominating over whatever hydrophilicity the $\ce{-OH}$ group had been providing for lower homologues.

From source [1]:


  .. any alcohol that is insoluble in the reagent might appear to be giving a positive 3° test.


The paper goes on to properly quantify several things:


  Based on the results of this study, using 1 drop of alcohol added to 10 drops of reagent, it can be said that, at the very least, all saturated acyclic monofunctional alcohols having six or fewer carbons are soluble in the Lucas reagent.


It is clear that the relative amounts of the sample and the reagent become important in such borderline cases (~6 carbon long chains). If too much alcohol is added, it may cause an illusion of positive lucas test.

From source [1]: (emphasis mine)


  That statement may not be true when 3 or 4 drops are used
  per 10 drops of reagent. In fact, sometimes when 10 drops of
  reagent are added to 3 or 4 drops of some of the alcohols, an
  interesting event occurs after a delay of a couple of minutes
  or so. Apparently, some undissolved alcohol that has been
  clinging to the bottom of the test tube suddenly breaks loose
  and, by streaming to the surface of the very dense Lucas
  reagent, gives a momentary illusion of a positive test. The
  real cloudiness, alkyl chloride formation, appears later—all
  the more reason to use a high reagent/alcohol ratio.


References:

[1] "A study of the Lucas test"
R. A. Kjonaas and B. A. Riedford
J. Chem. Ed. 1991 68 (8), 704
DOI: 10.1021/ed068p704