The condition you are looking for has a name: absolute isolatedness.
In fact, a surface singularity is called absolutely isolated if it can be resolved by using only quadratic transformations centered at the pointreduced points, that is, no normalizations will be required.
In general, isolated surface singularities are not absolutely isolated. But, for instance, rational singularities are so.
Googling "absolutely isolated $2$-dimensional singularities" you can find a lot of references. For example, Tyurina's paper Absolute isolatedness of rational singularities and triple rational points can be surely useful.