I am currently playing a 3rd level Wild Magic sorcerer, and its a lot of fun. I don't know how your friend manage to reach 7th level without ever rolling on the wild magic surge, because you can force it by using Tides of Chaos first, then cast any non-cantrip spell. If your DM refuses to let you roll in this situation, you shouldn't be playing a wild mage.
My favourite is using Tides of Chaos to grant advantage on Chromatic Orb, and twin it. Its a lot of power, but the limiting factor is your sorcery points, which I feel is too little, especially at low levels.
I had my doubts about Wild Magic at first, but its all gone now. Its fun, and thats what I want when playing D&D. Everyone always thinks about the Fireball TPK thing, but its only 2% chance. I seldom roll on the wild magic table at levels 1-2, because i hardly use Tides of Chaos. Most fights are easily won with Sleep anyway. Rolling dice is fun, and rolling a d100 is exciting. Unless for some reason you hate randomness (what the hell is wrong with you??), then Wild Magic is not for you.
Special note on Tides of Chaos: Its the greatest thing about Wild Magic. Having advantage on attack, ability checks and saving throw is good, but what makes it golden is the ability to constantly spam it. Using a spell scroll, death saves, etc all benefit greatly from Tides. Draconic resilience is not all that good, unless for some reason you are getting hit a lot.
Tl;dr - Draconic is not clearly better than Wild Magic until 14th level, which they get free flying movement.