Essentially it's emphasis, and while it is often used to express opinion, it isn't always. J-J dictionaries will give the definition as 徹底的 or something. So in general it's a very neutral word that I don't think changes the tone too much. Rather it's what it's connected to that determines the tone.
In the example that you give, I don't think that it necessarily makes the sentence more humble as much as it is trying to make oneself not appear arrogant or overly assertive. Maybe this is a small distinction, but given the nature of あくまで I don't think we can say it makes something more humble. Like you say, it's very similar to saying, in English, "It's only my opinion," and I think it carries the same nuance in this situation. That is, you're offering up an opinion that could potentially be controversial or seem overly assertive, so you try to distance yourself from that interpretation by emphasizing the fact that it is, in the end, just your personal opinion, and not necessarily the opinion.
Here are some examples from alc that show some of the different uses:
be all just superstition
This one is a more neutral one that just has the meaning of "to the end"
consistently maintain that
This one would be the opposite of what your example is, mostly because it's connected to 主張, an explicit statement of assertiveness and 'persistence.'
So to address your two points once again, in the context of your question, it has an effect like making the sentence more humble, but it can be the opposite as well depending on the context. With regard to persistence, it doesn't necessarily mean (in your example) that this person is strongly suggesting this or that it's a long-held belief or anything. It's just an emphasis on the fact that it is a personal opinion rather than fact.