Singularity is by definition one dimensional and thus inherently homogeneous.
It need not be true that the universe started from a point. There are solutions exist solutions to the Einstein's theory which allow a n infinite universe mand the universe is infinite from the initial instant onwards. This takes care of your objection. There are mechanisms which can generate imperfections or inhomogeneities without violating any know principle of Physics. Once the fluctuations have been generated, gravity makes sure that large structures can form.