It's the conclusion that this makes them 'arbitrary' that I disagree with
My reasoning is sort of, If there is no objective moral end, then no moral system is greater than another. Therefore we cannot fault a murderer for following her morals. If we evaluate her actions from the perspective of our morals, then even that evaluation is arbitrary in the general sense. Why would our criticism in accordance with our morals hold any more weight compared to a criticism in accordance with another system? From our moral standards, we could criticize Nazi Germany policies as bad, but does that criticism hold any weight? If moral ends are truly subjective, then we don't have a solid basis to reason that our criticisms of Nazi Germany policy are any greater than the policies themselves. That is why I say arbitrary.
Ill give an example to illustrate:
Imagine that we had an object (lets say a ball). Now, we also assume that there is no objective fact regarding the color of this ball (just as moral relativism sees no objective principle relating to the morality of actions). Person A, see the ball and says that it is red. Person B, sees the ball and says that is is blue. Now, no matter how much they arguer or criticize each other's position with regards to their own references, we cannot ever make a meaningful evaluation that Person B's claim is any way better or superior to Person A's (or vica versa) simply because of the fact that there ball does not have an objective color. In a similiar way, if we assumed that morality is truly relative, then we cannot meaningfully favor one society's morals over another.
There is no 'objective moral end'. There are only subjective morals. And yet people manage to measure moral systems against their various subjective moral 'truths' all the time.
If moral truths were subjective, it allows for evaluation against other subjective morals, but ultimately that evaluation itself is false, because the very values in question are subject to relativity. Arbitrary morals and subjective morals go hand in hand.
Only if your definition of 'progress' is limited to 'progress towards the objective finish line'.
Progress is not simply change. It is directional change. It implies a goal. There must be an end towards which progress is directed. If that end is subjective, then it is arbitrary.
I'll illustrate the example with the ball scenario. Person A and Person B, now decide to work together and try and find out the actual color of the ball (but remember there is no actual objective color). They perform a series of tests, observations, and now they see the Ball as green. Could we say that they have made progress regarding the color of the ball? No, because the ball does not have an objective color. Only if the ball did have an objective color (lets say green), could be evaluate the efforts of Persons A and B and label it as progress. Progress implies we are moving towards an end which exists as an objective standard.