Some people think that adding a core to a transformer winding will couple the windings closer together, reducing the leakage between them. Others have heard that the core will couple with the fields between the windings, dramatically increasing the leakage. The fact is that the leakage inductance of two concentric windings designed for a modern switching power supply transformer is almost independent of whether the core is present or not. Try the experiment if you don't believe it - measure the leakage inductance of two windings with and without the core present. If you're not from the US, don't even bother calculating what a circular mil is.
Current density is irrelevant.
What matters is how much dissipation is in the wire, and whether there is sufficient cooling to keep its temperature in the acceptable range. Given the extreme types of cooling which may be used in power supplies, ranging from liquid immersion at one extreme, to operation in the vacuum of space at another, the appropriate current density in the wire will vary widely. After designing practical transformers at Ridley Engineering for more than 20 years, frankly we have no idea what the current density of any them was. This myth results from an artificial constraint placed on transformer design to try and simplify and reduce the number of variables available. The losses might be similar in some cases, but don't worry if one is much lower than the other.