[RPG] What does carrying capacity do in combat

For reference see PHB, page 176. That's where most of the relevant information about carrying capacity is found.

Carrying Capacity. Your carrying capacity is your Strength score multiplied by 15. This is the weight (in pounds) that you can carry, which is high enough that most characters don’t usually have to worry about it.

Push, Drag, or Lift. You can push, drag, or lift a weight in pounds up to twice your carrying capacity (or 30 times your Strength score). While pushing or dragging weight in excess of your carrying capacity, your speed drops to 5 feet.

So no, as per RAW, you don't get advantage on picking up things or grappling. In fact, grappling is not affected by your carrying capacity at all, although they are both strength-based and will, therefore, be roughly proportional to each other.

What a high carrying capacity does allow you to do is being able to lift heavier objects at all (including corpses, which are objects), or at least without a reduction to your movement speed.

For example, if you are a half-orc with a strength score of 18, you have a carrying capacity of 270lbs and a maximum weight of 540lbs for pushing/dragging/lifting.

Therefore, if this half-orc would want to drag a dead creature, for example, with a weight of 200 pounds (which is below his carrying capacity) from A to B, he could do it using his normal speed. The wizard with a strength score of 8 (= 120 / 240 lbs) could do it with a 5 feet movement speed only, since it exceeds his carrying capacity, but not twice his carrying capacity.
Hence, while the half-orc could drag a 500-pound-corpse at a speed of 5 feet, the wizard would be completely unable to drag the corpse.

tl;dr A high carrying capacity does not give benefits to grappling, but it does increase the weight limit for pushing, dragging etc. beyond which you are slowed to 5ft of movement. It also increases the maximum weights that you can carry or drag at all.