Probably the biggest disadvantage is the loss of turns due to the insulation on the stranded wire which just takes up space. Magnet wire used to wind transformers and inductors has an enamal coating on the solid wire so you get maximum turns for a given diameter. It is able to withstand higher temperature than the insulation found on hookup wire.
Other than that you can probably use it without any ill effects.
Well, they do make stranded magnet wire, are you referring to that? This is known as "Litz" wire and is very useful for higher frequency transformers used in switching power supplies.
Due to the skin effect, you have much better current density under HF switching for small combined strands. This is necessary for maximum performance when designing transformers for flyback converters, etc.
If you are planning to push power (volts times amps) through the transformer, then solid conductors are to be preferred. High frequencies like lots of surface area, high currents at low frequencies like lots of circular mills.
Do you really need 6 gauge wire? 12 is pretty bad to snake around turns. I recall seeing that one outfit had got around the stiffness problem by rolling the wire out to a tape - preserved the current carrying capacity but made it easier to wind.
Turn-to-turn voltage isn't going to be much. You might be able to get away with using bare wire with a paper layer insulation, sprayed with lacquer to improve insulation and seal out moisture.