The Like and wildcard expression is the easiest to use if you want to match a given portion of a set string. However, there is a way to write code that is like your original approach, but the expression is not LEFT, it is SUBSTRING (at least for file geodatabases and shapefiles). The basic syntax is:
SUBSTRING(string_exp FROM start FOR length)
So for your problem it would be:
SUBSTRING(FEATURE_TYPE FROM 1 FOR 1) = 'Y'
Your title suggested you wanted to find any string that began with X, Y or Z. That could be done with this expression:
SUBSTRING(FEATURE_TYPE FROM 1 FOR 1) IN ('X', 'Y', 'Z')