The coefficient for the highest term in a polynomial should not actually be non zero as this results in an imperfect Shamir scheme.
This is because this assumption means that any attacker will know that the polynomial is definitely of a size t-1. Now if said attacker gained t-1 shares it is actually possible for him to compute a number S' that is not equal to S (the secret)
now although this doesn't seem like much it actually means that the scheme is not secure as the attacker definitely knows that S' is not the secret (as S' is not equal to S). Hence all coefficients should be chosen randomly and uniformly as even if the attacker has t-1 shares of a Shamir scheme in which the highest polynomial's coefficient is 0 there is now way (in an information theoretical security perspective) for him to know that he has computed the secret.
this is explained and proved in more detail on page 4 of the following paper: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.44.6353&rep=rep1&type=pdf
In relation to your question i do not believe that some of the coefficients being equal to 0 will result in any more or less work for an attacker as hwo are they going to know which coefficients are non zero or zero?
hope this helps :)