This is incorrect, you can put non-zero divergence of magnetic field and all the equations and predictions stays the same. Better it would make Maxwell's equations symmetric under exchange of fields and sources. \

so close to zero that monopoles would be extraordinarily weak if they did exist

Why? I can't think of a reason why would this be the case? You are not solving Maxwell's equation for the universe. You can have divergence of electric field closed to zero because you have very low density (the field source) in the region you are studying.