Well, the beginning starts with understanding incest biologically. Biologically, we are talking about organisms with the sexual reproduction mode, and there we care about the terms inbreeding and outbreeding. In nature, both are common, and both are functional evolutionary strategies. Genetic traits can be roughly categorized into recessive and dominant traits. When genetically different individuals mate, the chance of at least one of the two having a dominant allelle for a trait is relatively high. In contrast, when two individuals have comparatively similar traits (ie. they have high “homozygosity”), then it is likely that if one has a recessive allele for a trait, the other will too. This can lead to what is called “Inbreeding Depression”. If such recessive traits are, as they call it, “deleterious”, this tends to end up as a selective reason for individuals to engage in outbreeding instead of inbreeding.

However, this isn't always the case and the opposite can be true. “Outbreeding depression” is essentially the same as inbreeding depression, but works the other way around. Inbreeding is commonly used in domestication to eliminate deleterious traits and establish new, desirable traits. In general, inbreeding increases gene expression, ie. it increases the chance of both beneficial and deleterious recessive traits getting expressed, which either increases or decreases the fitness of the offspring.

Increased homozygosity becomes an issue when there are deleterious recessive alleles in the gene pool of a family. For humans in particular, inbreeding can generally be considered harmful for the child, as it substantially increases the risk of expressing deleterious recessive genes. This is especially problematic in small populations where genetic variation is already limited. Now, this isn't said and done, as it is entirely possible for inbreeding to lead to the expression of beneficial traits as well. As far as the resulting offspring is concerned, this is on a case by case basis. For the parents however, especially if they don't have a genetic study done, this is not a reasonable risk and I believe it it justified to say that humans should abstain from inbreeding (procreative incest) for medical reasons.

This medical reason, however, begins to point us into an important moral direction - there is a difference between incest and inbreeding. Inbreeding is a problem associated with genetic problems in the offspring. However, while inbreeding can be part of incest, it doesn't have to be. As RationalWiki puts it:

On its own, the medical reasons against inbreeding would give rise to a crime like “sexual recklessness”, if you want to call it that way, but that is only a good reason against inbreeding, not all kinds of incest.

Another interesting angle on incest is the problem of power dynamics. Parents and their children, even if both are above the age of consent, have an inherent power dynamic that encroaches on the ability of the child to actually give consent. This pervades across the hierarchy as everyone lower on the family tree answers, in one way or another, to those higher on the family tree. For simplicity sake, I will refer to an individual's height in the family tree as a “Rank”.

Now, I will conceptually agree with this argument, but I would like to point out that this line of argumentation is risky, because it assumes that one cares about family structures. Family is not real, family is a social construct. To me, my family is literally just another person on the street, I wouldn't even recognize half of them if I met them, their position in a family tree does not give them any authority over me whatsoever. This may have been different with my parents when I was under 18 years old, but now that I am of the age of consent it becomes really easy to recognize that family is literally just a brain thing that you either agree with or don't. But, it doesn't take reaching the age of consent to realize the electability of the family construct.

The “family” is likely anchored into your nation's laws, but that is not a prescription for individuals to accept families on a personal level. In law, families are generally just groups of sexually related individuals, but in the end they're all just individual humans. And, as an example, any individual human can be capable of taking the custody of a child, sexually related or not (in fact, we already have this with foster parents and the such). What I'm saying here is that families exist in the sense that they are used to determine roles and obligations (ie. parents care for their children), but these are constructs we created and which no individual is forced to believe in. If you don't want to think of your brother as your brother but merely as a person born by a biological relative, then that's entirely up to you to do. Everything else you may or may not feel about your brother is the result of you, as an individual, thinking of him as part of a social construct you allowed yourself to believe in. And while that is perfectly valid, rejecting these constructs is equally as valid. One isn't better than another.

The only really good argument here is ranks and power dynamics. Your parents hold power over you as the child. This imbalance creates inherent problems with your decision-making. This is, and there is no doubt about that, a serious problem if the child is still underage. Sexual relations between someone who is underage and someone who is of age is unethical and under no circumstances acceptable¹⁾. This is already different if the child is of the age of consent itself, or if the potential incest is horizontally between two individuals of the same rank, ie. brothers/sisters, cousin, cousin-in-law (lol?) (but not vertically between parent/child, grandparent/grandchild etc.).

Between brothers/sisters/cousins/cousins-in-law, no such power dynamics exist. For where these power dynamics do exist, bringing up incest as a problem is a reasonable objection. Personally, I care more about the potential age difference than about whether it's the same family - both must be of the age of consent or - if underage - roughly the same age. If this condition is satisfied, I find “family” to be a weak argument against it, unless of course the goal is to have procreative sex, see 1. Genetics.

Others put forward that incest is “unnatural”, but this argument runs into the same problem as the “it's disgusting” line does: It's a Value Judgement. Why is it unnatural? Who decides what is or is not natural? Why does being counter-intuitive make it unnatural? What's bad about doing unnatural things? Does it harm or hurt anyone? I have yet to see an argumentum ad nature that isn't just purely based on intuition.

There are so many problems with this. Which god? Why that one? How can we verify our chosen god is real? If those three questions can be answered, then next why care about what a god has to say? Gods are an opinion and people are entitled to theirs, but whether something is “good” or “bad” is an empirical and philosophical question with near-absolute answers, upon which god only has a bearing if we know or at least have reasonable grounds to believe that one exists.

Actually a good point. The age of consent for incest is definitely something that would require some attention. We could even agree that it should be higher than that for normal sex, but it should be reasonable. Maybe 18+? I think the biggest source of damage to the livelihood of incestuous people comes from societal stigma, not incest itself. Ethically this may be considered flawed, but either way this means that falling such decisions²⁾ requires some additional maturity.

I actually don't even know here. It might be. Is incest related to developmental disorders? Does removing the prohibition on incest promote those disorders? I mean, I don't know why people would do incest in the first place, I've never talked to one about their motivations. I could understand younger people, especially children who don't really make these kinds of distinctions (but then also run into the issue of being unable to consent), kind of just experimenting around; and their family members, especially siblings, will be the easiest test subjects to come by. Look, I don't know when, why and how that kind of stuff happens, I'm really just pulling things out of my ass right now.
Anyway, seeing how counter-intuitive and disgusted even I feel thinking about incest with what little family I have, I can't rule it out that there could be psychological issues attached to the emergence of incestuous behaviors. Not causative factors per se, if anything then probably just correlation, but maybe just nothing at all. But then again, on a large enough population, all kinds of things are mathematically bound to happen eventually, so in case it is strongly correlated to developmental disorders, there would still be more than enough circumstances in which it would just happen. Just like that.

The relationship between a parent and their child is special. It is based on safety and mutual trust. It feels like a parent engaging in sexual activity with their child is violating those principles of trust and compromises the safety of that child.³⁾ However, while I can agree that it really does kind of feel that way, I can't really explain it. Why would it compromise the safety of the child? Why would it break their trust? Because I set different expectations for myself? Who decides that the line of trust ends before and not behind incest?