The problem is that where $a_1(x)=0$, your equation is not really second order. It doesn't mean that it doesn't have solutions, but fact is that the zeroes of $a_1$ are singular points, i.e., points where usually bad things happen to the solution.
In most cases, $a_1$ has just a few zeroes, and one simply works within the intervals determined by those zeros. Within those intervals, dividing by $a_1$ is fine.
As you say, if your goal is to solve by using power series, dividing by $a_1$ may complicate things. Note, though, that if you will have $a_1$ as a nontrivial power series, you will need to multiply that series with that of $y''$, and in most cases you won't get anything remotely pretty. The most common cases occur when $a_1$ is a power of $x$ or another equally simple expression.
If you are using a method other than power series, like variation of parameters, most of the time the method is phrased with the equation in a form where the first term is $y''$ alone.
Finally, to answer your title question: in general you will not lose solutions because the singular points are isolated and you look for solutions on intervals. If $a_1$ were to be zero on an interval (in which case it cannot be analytic), then you would indeed lose solutions by dividing. All methods I know wouldn't apply in that situation, though.