[RPG] Are Goliaths able to wield large-sized weapons

As RAW, no, it does not

Being considered a size higher for carrying capacity and push, drag, or lift force is not the same as being a size higher for all purposes, including grappling.

You count as one size larger when determining your carrying capacity and the weight you can push, drag, or lift.

As RAI, maybe

The loop hole is that the PHB has no reference on why and how the grapple-er can move without speed penalty if it is two size higher, and the DMG does not touch the subject. There are two potential scenarios for why a grapple-er one size or lower has it speed reduced, though; the grappled creature have enough leverage to make the movement difficult or that the grapple-er cannot carry comfortable enough the grappled creature as in "difficult terrain", and that the weight of the creature is the deterrent.

In the first case, since Powerful Build does not increase per-se the size, it is safe to assume that it does not help to increase the comfort or reduce the leverage and, such, the speed is halved.

In the second case, where is the weight that maters, things change. Powerful Build improves the carrying, dragging, and lift capacity, therefore the grapple-er creature does not have its speed halved.

Which one it is

The only evidence I found is in favor of is that weight is the reason for the reduction in speed. The first part is in the text of moving a grappled creature (emphasis mine).

When you move, you can drag or carry the grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you.

As far as I can tell, there are no rules on how you drag or, particularly, carry a grappled creature. You can carry over you head a grapple creature as if it were a sack of potatoes. In PHB 176 there is a segment that shows how can you drag, carry and lift something in particular and, as shown, it is a STR and weight contest in which Powerful Build should work as intended.

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).

Bottom line

At the end is the DM that has the last word. Talk to your DM and present the evidence. If you are the DM decide what it is best for your campaign. If a particular decision is more powerful that you expected it is in your right to take back the decision.

Relevant spell RAW is:

An enclosed creature can use its action to push against the sphere's walls and thus roll the Sphere at up to half the creature's speed. Similarly, the globe can be picked up and moved by other creatures.

Relevant phb RAW is on pg 193:

Your character can do things not covered by the actions in this chapter, such as breaking down doors

When you describe an action not detailed elsewhere in the rules, the DM tells you weather the action is possible and what kind of roll you need to make, if any

So the sphere can be moved by pushing. A door or other movable object could also be moved by pushing. If on the other side of the door, or sphere in this case, is an enemy, you're pushing against an active agent with a strength or athletics score. Just like in a friendly game of tug of war, it's your check against theirs.

If the enemy in the sphere is incapacitated, then you're pushing against a lump of mass. Then the sphere is a convenient protected cart of sorts. That's when your push/pull/drag rules are going to come into play.