I don't know what Hintikka said about this. However, note that Hintikka argues against intuitions as an alleged source of evidence, which is their primary use in contemporary philosophy. As far as I can see, axioms aren't considered evidence. They are postulated. I think physicists are more careful in this regard because they explicitly speak about postulates, not axioms.

My take is that most texts in which intuitions are applied as a source of evidence (as opposed to an indicator for further inquiry) can safely remove any talk about intuitions and nothing of value and philosophical insight is lost.

I recommend recommend Hintikka's The Emperor's New Intuitions (1999) for arguments why intuitions as a source of evidence are useless and a huge scandal in philosophy. Intuitions used to be a methodological shortcut in natural language syntax because native speakers have (to a certain extent, which used to be a bit exaggerated) a capacity to judge the grammaticality of sentences even in light of frequent performance errors. There is no evidence of such a capacity about philosophical matters. To be honest, I doubt a single "datum" could be discerned in philosophy that is established by intuitions about which other philosophers don't have intuitions to the contrary.

The use of "intuition" in contemporary philosophy illustrates the lack of a commonly agreed upon methodology and subject matter.

My view so far is roughly speaking naturalist/pragmatic in the sense that we figure out which values are truly objective in the same way as we would figure this out about any other issue. Maybe it's more about the stance towards a specific value. We sometimes speak of specific values in strongly realist ways. Whether we're right or wrong about this is a matter of nature, and, if you want to put it in these terms, concerns value epistemology. Although I'm sympathetic with error theory as a critique, I find it overall not very credible as a statement about all value.

Francesco Orsi makes similar distinctions as I do. Orsi (2015): Value Theory. Bloomsbury Academic. It gives some taxonomy, otherwise I don't think it's very interesting. Here is my take, but whether you think these are examples of the types of values really depends on your metaethical stances:

+ subjective and agent-relative:

John: Becoming a lawyer is better for me than becoming a clarinetist. I appreciate the better job prospects and payment in the long run and I'm not really sure I would continue to like music if I became a professional musician.

John: Bungee jumping is great, it's better than going on a hike.

+ objective and agent-relative:

It is better for John not to become a lawyer and pursue a career as a clarinetist. John would probably not even pass the bar exams and the profession would invariably burn him out. He doesn't have the personality for it.

Bungee jumping is horrible for John. Because of some traumatic experience as a child, he's afraid of heights. However, he isn't aware of that yet.

+ subjective and agent independent:

John: Basic democracy is better than representational democracy. At least, I'd much prefer to live in a direct basic democracy.

+ objective and agent independent:

For various reasons <long list of reasons>, basic democracy is vulnerable to disruption and representational democracy is much better.

I understand the motivation and what you say but can't agree. IMHO the object/subjective distinction is indispensable, without it you cannot understand metaethical positions that talk about agent-relative, objective values. For example, if someone is an extrovert and likes meeting people, then it can be objectively good for them to have the opportunity to do so (notwithstanding conflicts with other values or reasons against it). But the value is clearly agent-relative, since there are also introverts with other needs. Many human needs are like that.

The attribute "subjective" is a bit special in my opinion, however, since it is better to speak of personal preferences instead of subjective values. These reflect what people like and what they prefer at a time. We do not generally talk about these as values, although sometimes we'd call some of them "personal values."

Orthogonal to these distinctions is in my opinion the agent-relative vs. agent-independent axis. Basically any kind of value that concerns a human is agent-relative.

To cut a long story short, agent-relative vs. agent-independent is one axis, subjective vs. object another axis, and absolute vs. relative yet another axis. I accept all distinctions, it's just is part of my personal view that talking about subjective values is a bit of a misnomer.

I think you might be right, I wrote from memory and would have to re-read the relevant passages. I've always been skeptic of intrinsic values, so maybe I too easily thought Scanlon provides an example of it. Scanlon discusses the example primarily as an example against consequentialism, though, which is another matter.

But there might be another terminological misunderstanding here because in my opinion the question whether there are intrinsic values or not is irrelevant to the value realism I hinted at. There can objective (or, at least intersubjective) agent-relative values regardless of whether they count as intrinsic or not. For instance, you can be a utilitarian and consider utility a type of agent-relative, objective value. What is best for each person depends on that person in this view, yet it's not a subjective matter what is best and people are frequently wrong about it.

Intrinsic? Friendship would be an example. Scanlon devotes a chapter to this topic in his "What We Owe to Each Other." Anyway, value realists do not even need to assume that some values are intrinsic. Which values are intersubjective or even objective depends on whether there are facts that make corresponding value statements true and on the nature of these facts. Such facts could be moral facts, but they could also be aesthetic or prudential, for example, or there could be a naturalist position that grounds them in empirical facts about humans in general. It really depends on the kind of value theory you think is the right one.

As a metaethicist I'm undecided about this / don't think there is a clear solution. IMHO, claiming there are moral or aesthetic facts is in many ways problematic and dubious.

I'm fine with many things the author states but find the idea that axiology is not part of ethics somewhat counter-productive. In my opinion, the deontic realm the author talks about is one branch of ethics and axiology is another branch of ethics. That is also how it has been conceived traditionally. That the author wants to banish all axiology from ethics seems to be based on the idea that values are somehow personal and agent-relative from the start, but only few people would subscribe to this view. Even utilitarians have proposed more elaborate views of value, for example Harsanyi takes informed preferences under some veil of ignorance construction as the basis of his theory. I'm not a utilitarian, but I would defend axiology as an independent and inherently normative part of ethics that is not reducible to deontology (or vice versa).

To make this clear, however, I think that only part of value theory belongs to ethics. There are plenty of values that are not morally relevant.