That place is called Saahil, where the waves crash with a huge bang but only during high tide. It is on the same Beach Avenue, just after Village restaurant on main sea view beach. If you're coming from Clifton (McDonalds, Sea View), go straight, or if you're coming from Khayaban e Ittehad, turn left on Beach Avenue. You can easily do it along with a trip to Do Darya restaurants, which are further down the same road.
For BBQ, you will have the option from BBQ Tonight (Clifton), Waheed (Burns Road), and Ghaffar (behind Tariq Road). You will have plenty of food options other than BBQ, like Haleem (Samn abad), Nihari, varied types of Fish (Yaadgar at Jamshed Road, Kaemari sea food street), different types of Biryani (Student Biryani at Saddar, Food Center at SIndhi Muslim/26th Street, Biryani Wala at Boat Basin, Ghausia at Liaqatabad). Just ask the waiter to put in less spicy rice and they will do it easily. The art of Biryani is not just in cooking, but in mixing also. Hyderabadi (Deccan) biryani is also available at Hyederabad colony in front of Central Jail, which is very close to Yaadgar fish. I must say that if you have the option, try Beef Biryani beacuse to me, chicken doesn't give that flavor which Beef has.
As for snacks or munching, you can try Lachay wala paratha, aalo bhara paratha, chicken & cheese paratha at Quetta Alamgir hotel near Bahadurabad, which also has Tosso at Bahadurabad round-about for halwa puri in the morning, Dossa from a make-shift suzuki pick up kitchen (only evening), ice cream and falooda from havmore and others at the other side of the round-about, or Gola Ganda + Gol gappy and chat from nearby Dhoraji.
In short, you will have the following so-called food streets/areas:
Gulshan e Iqbal (Disco Bakery road, all types of restaurants)
Gulistan e Jauhar (Balochistan Sajji restaurant has variety of food)
Hussain Abad (many eateries)
Bahadur Abad and adjoining area (varied types of food)
Sindhi Muslim roundabout (up scale to mid range restaurants)
Landi Kotal chowrangi, North Nazimabad (Donisl Restaurant road, mid range restaurants)
Do Darya (mostly upscale restaurants, my favorite is Kolachi)
Boat Basin (varied types of food)
Burns Road (all types of food)
Port Grand
Kaemari (primarily for fish and prawns)
Highway restaurants (for karahee or haandi). There are far too many good ones, but my favorite are Madina which is 5km ahead of Bahria Town Karachi for Balochi Karahee Tikka, and Usmania which is adjacent to Bahria Town for its chicken chatpata and fish.