Those aren't rules in conflict. They're overlapping effects on the PC.
If the PC is physically capable of dealing with the weight then they are not reduced to 5' movement. If they are not then the excessive weight causes them to have a maximum movement of 5'.
That places medium sized characters into the position where from 8-20 STR they can carry 120-300 lbs and they can drag 240-600 lbs. It only takes 8 STR to drag 240 lbs, which seems capable of dragging a similar sized creature even for someone with under avg STR.
While dragging, lifting, or pushing the weight above carrying capacity the PC's movement cannot be more than 5'. If they are dragging within that carrying capacity then the grappled rule applies.
Moving a grappled opponent doesn't change anything from the carrying capacity rules. All it does is cost extra movement.
The Grappler feat doesn't change anything in the carrying capacity rules. All it does is remove the extra movement cost when moving the grappled opponent. All these rules can be applied at once.
If you make a Goliath the Powerful Build option helps this significantly, and if your party cleric is a Halfling then you're Goliath monk can happily run around carrying the cleric spreading Spirit Guardians love. But there's no rules conflict going on here.
Breaking those limits would require a DM to set a DC Athletics check in a "DM may I?" approach.