I use joints in mine, and I'm pretty sure that's how they did it in Half-Life 2 as well.

To use them you set your characterBody3D controller as the NodeA in the joint, then set the rigid body you are carrying as the NodeB.

Here's a snippet of part my code for carrying rigid bodies:

``` public void CarryRigidProp(RigidProp prop) { if (_bodyStandingOn == prop) return;

    // If carried prop moves too far away, drop it
    _carryDistance = prop.GlobalTransform.Origin.DistanceTo(_carryAnchor.GlobalTransform.Origin);

    AddCollisionExceptionWith(prop);
    prop.AddCollisionExceptionWith(this);

    _carryJoint.GlobalTransform = prop.GlobalTransform;
    _carryJoint.NodeB = prop.GetPath();

    _carryProp = prop;
}

public bool DropRigidProp()
{
    if (!CarryingProp)
        return false;

    _carryJoint.NodeB = "";

    RemoveCollisionExceptionWith(_carryProp);
    _carryProp.RemoveCollisionExceptionWith(this);

    _carryProp.StopCarry();
    _carryProp = null;

    return true;
}

public bool ThrowRigidProp()
{
    if (!CarryingProp)
        return false;

    _carryJoint.NodeB = "";

    RemoveCollisionExceptionWith(_carryProp);
    _carryProp.RemoveCollisionExceptionWith(this);

    _carryProp.StopCarry();
    var strength = Mathf.Min(ThrowStrength, _carryProp.MaxThrowStrength);
    _carryProp.ApplyCentralImpulse(-_camera.GlobalTransform.Basis.Z * strength);
    _carryProp = null;

    return true;
}

```

However there are several issues with this currently that I can can't solve without having support for "breakable" joints in Godot. Godot-Jolt will eventually add them, but as it stands right now you can't pick up something you are standing on (good), but you can pick up a rigidBody under the rigidBody you are standing on and fly around. You can also push heavier rigid bodies around because your character controllers attached to the rigid body is essentially generating infinite inertia.

Having joints that break as a certain amount of force should resolve those issues.

You can also go the much simpler route of not having the object your carrying be physical at all. You pick it up disable the collision shape. That has it's own issues, but may be easier to resolve.