This is my point. You are both wrongly assuming that self defence will bear any resemblence to what happens inside the dojo, and that you wil be "fighting" a skilled & trained martial artist. Do people regularly get into street brawls with other trained martial artist, it happens yes, but martial artists (At least the ones I know) are not in the habit of brawing in the street.
Do people think criminals spend time training in martial arts? They don’t "fight", in fact a "fight" is the last thing they want, as such becoming an "all round martial artist" is not necessary for dealing with career criminals. In The Complete Book Of Urban Combatives, Lee Morrison says "The physical side of the equation is pretty simple: just develop two or three effective strikes that can be executed hard and that work well."
You only need to become an "all round martial artist" for the purposes of self defence if you a) are martial artist who wronly assumes you will be gettign into street "fights" with other skilled & trained martial artsist, or or b) you do not understand the realities of criminal violence, or the "rituals of violent" used by professional criminals.