Why can't I get that grilled taste from my outdoor grill? We purchased a gas grill years ago. We have caste iron grates and high to low flames adjustment. It has this metal plate that covers the flames. I guess it's there to prevent grease from hitting the flames.

Whenever we grill burgers, chicken or steak, they don't have that characteristic grilled taste you get from a restaurant. The grilled taste I am looking for is that charred taste, which is the same taste you get from a good steakhouse or shish kabob. It's the same charred taste you get from the Whopper or a really good burger cooked over open flames. It's not smokey as in BBQ using wood chips.  

I'm not using lava rocks, could that be an explanation?
Based on your updated question, the major factor is that your fire simply is not hot enough to produce the charring and browning reactions you desire.

A steakhouse salamander (think of a broiler&mdash;grill in British parlance&mdash;which has flames on both the top and the bottom, to cook both sides at once) can be up to to 1000&deg;F and the elements are mere inches from the steak on both sides.

Gas grills have a maximum amount of heat they can produce.  The only thing you (may) be able to adjust is the distance to the flame.  The major cooking modality in a grill is infrared thermal radiation.   This decreases per the square of the distance, so the closer your food is to the burner, the faster it will cook (within reasonable limits).

Other than that, you may need new equipment if this is in fact the root cause of your issue.