The main reason to go out to eat on dates is that you actually communicate with your...erm...partner. Most other dates which involve activities such as going to the movies keep things require divided attention and therefore cause disinterest or lack of interaction with one another. During "dinner dates" people can also learn pieces of the other person's personality, such as their amount of generosity, taste, and politeness. Not to mention GIRLS and GUYS LOVE FOOD. Basically to summarize one goes on a eating date to communicate, learn, enjoy oneself (even if your date sucks you still get food), and to decide if you are for or against being with this person. Not to mention it is the safest(although most uncreative)option for having a decent first date or really any date.
Note: You may want to try to get over that phobia or whatever...its not going to benefit your love life and may make you seem snobbish or egotistical. I know eating isn't amazingly attractive, but don't let that affect you so much.