If my having done (or having failed to do) X is a sin, then my having done (or having failed to do) X is something for which I am morally responsible. But I am morally responsible for an act or omission only if I could have done otherwise. But if I could have done otherwise, then it cannot be essential to me (part of my nature as a human being) that I sin (or be in a sinful condition, or be guilty). Whatever guilt accrued to someone in the past (Adam or anyone else) in virtue of his misdeeds is his affair alone and is not chargeable to my moral bank account.
As he goes on to note, you can get around this by saying that original sin is not actually a sin:
OS is not, strictly speaking, a sin but refers to a sort of structural flaw or weakness, one to be found in each and every human being, which predisposes us to actual sin but is not itself a sin or a state of sinfulness for a postlapsarian man or woman. This predisposition might be ascribed to the hebetude of the flesh or the inertia of nature. Whatever its source, it is not in our power. Hence we are not responsible for it and not guilty in virtue of it. It does not interfere with our free will or make impossible self-perfection. There is no inherited guilt. Perhaps the structural flaw under which we all labor is the result of someone's sin in the past; but if it is we are not morally responsible for it.
There are different accounts of original sin, but taking original sin to be a predisposing flaw or weakness is precisely the route most people have taken, and is one of the marks of the family of accounts that has the most longstanding reputation for orthodoxy. Aquinas's position, for instance, is that original sin is only sin in you or me in the sense that sin is found in a hand or a foot -- it's not the hand's sin or the foot's sin, although it can be referred to the hand or foot in the sense that the hand or foot participates in the action: "just as the actual sin that is committed by a member of the body, is not the sin of that member, except inasmuch as that member is a part of the man, for which reason it is called a 'human sin'; so original sin is not the sin of this person, except inasmuch as this person receives his nature from his first parent, for which reason it is called the 'sin of nature'" (ST 2-1.81.1).
It does not follow, however, that we are not morally responsible for it, because there is more than one kind of moral responsibility. There is a perfectly reasonable sense of moral responsibility in which you can be morally responsible for people under your care or general supervision, for instance, even though they are their own agents, and even though in a stricter sense of 'morally responsible' only they are morally responsible for their actions. A saintly parent may have children who are vicious for no reason that is the parent's fault; the parent is not guilty of the children's bad actions, but as parent still has a moral responsibility for them -- to attempt to rein them in, for instance, to help correct harms they cause to others, and so forth. And we find this elsewhere in families, friendships, close-knit communities, and even societies as diverse and diffuse as religions and nations. Let's call this the moral responsibility of solidarity, to distinguish it from the kind of moral responsibility Bill has in mind, the moral responsibility of agency.
I can think of at least one more kind of thing that has perfectly good claim to being called moral responsibility, and that is the kind of moral responsibility we have given the things that we receive from others: that is to say, things we have to care for or (alternatively) look out for precisely because we are not hermetically sealed singular individuals but people whose history includes more than ourselves. If you receive a patrimony, you have a responsibility for it, not merely because you have a general responsibility for your own actions, but precisely because it is something you have received to be responsible for. And this includes negative features of patrimony, too, like debts and feuds, which at the very least are things one must deal with precisely because you have inherited them, even if you had nothing whatsoever to do with the actions that built them up and even if the people who did so where not under your care in any way, shape, or form. It need not, of course, be a matter of inheritance in the strict and literal sense; education and tradition and concomitant consequences can be the source of such responsibilities as well. It is, I think, in precisely this sense that Socrates held that it would be irresponsible for him to flee the laws of Athens and it is in this sense that people rightly do not lightly give up the customs and practices of their ancestors. Likewise, it need not be external. It is in this sense in which you are responsible for a bad temper inherited from your father, or for climbing your way out of a bad upbringing, namely, that even though you were not the responsible agent, it has come to you, anyway, and you have to handle it well. Call this the moral responsibility of legacy.
Clearly there are all sorts of relations among these kinds of moral responsibility, but I think we have to say that in themselves they are simply very different. Certainly moral responsibility of agency is very different from moral responsibility of solidarity or legacy, and it seems to me that all three have rather different kinds of objects. Likewise, these are all reasonably called moral responsibility; indeed, I think we probably more often talk about responsibility in the solidarity or legacy sense than in the agency sense: we are responsible for our property, our family, our friends, our nation, our church, etc., even though (in general) none of these are themselves things for which we can be blamed or praised.
When it comes to original sin, the common view is that we have no moral responsibility in the agency sense, but we are morally responsible for it in the legacy sense and in the solidarity sense. (I think it can be fairly easily argued that this is the view of most and perhaps all of the major scholastics, for instance, and many other theologians as well, as well.) One of the common ways in which accounts of original sin vary is how much emphasis they put on the legacy sense versus how much emphasis they put on the solidarity sense. You do find accounts that tend to accept only the legacy sense or only the solidarity sense; they are usually considered either only marginally orthodox or outright heterodox, but details can matter a great deal on this point. Most of the robust accounts clearly imply both, although they don't necessarily emphasize them equally, and all of the main family of accounts deny that we are morally responsible for original sin in the agency sense.
Which brings us to the question of guilt. Guilt in our usual sense is obviously purely a matter of moral responsibility of agency: it is precisely one of the clear distinguishing features of it, in fact. There is a view, called inherited guilt, that is often associated with the doctrine of original sin, although not all accounts imply it. It's a bit tricky to pin down what the view is, or to determine which accounts actually imply it; the most plausible and developed accounts that regularly use the term are generally Calvinist and are accounts of guilt by imputation. Going into imputation gets into all sorts of potentially complicated issues that I can't address here. The common source for such views is Augustine, and it is quite commonly attributed to him. I think it is somewhat misleading to do so; he does hold that we inherit what he calls reatus. This can be translated as 'guilt', but the term is much broader than what we usually mean by guilt, since originally it could mean a liability of any sort. The same is true for other well-known inherited guilt passages, like the Council of Trent's talk of the guilt of original sin; the scholastics, in fact, had regularly distinguished reatus in the sense of fault (liability as in 'at fault') from reatus in the sense of having to penalty (liability as in 'having to pay the consequences'); obviously the two can come apart in various ways. Now, while agency-liability (such as guilt) obviously only arises under moral responsibility of agency, it is, I think, a reasonable question as to what sorts of liabilities may arise under moral responsibility of solidarity and moral responsibility of legacy. That such liabilities do arise is quite clear: parents, for instance, often get them from misdeeds and mistakes committed by their children through no fault of the parents themselves.
One thing I find a little puzzling is that Bill calls what is the common view a 'Pelagian escape route'. I think Pelagianism is a different issue: it's really a position about the way in which we need God's grace. What connects it to original sin is that original sin implies certain things about our need for grace, and you can make Augustinian arguments that Pelagianism fails to do justice to this need for grace. The flow, in other words, is from original sin to a particular kind of need for grace to rejection of Pelagianism as failing to recognize this need properly; or, by modus tollens if you are a Pelagian, from Pelagianism, to a rejection of our need for grace in that particular way, to a rejection of original sin. But the positions are somewhat removed from each other. And accounts that deny personal responsibility and guilt for original sin likewise do not imply Pelagianism as long as they affirm the kinds of responsibility and liability that lead us to recognize the relevant kind of need for grace. I think it's clear enough that the predisposition accounts, which, after all, are the majority of major views, have long been accepted as being able to do this.