The argument here doesn’t require making any claims about the permissibility of euthanasia in general. Rather, I’m inclined to take as my starting-point something much more syllogistic: that if euthanasia is going to be allowed, we should be prepared to allow it for children. The reason for this is pretty straightforwardly an anti-ageism appeal. To have a law that prevents someone from accessing assisted dying of whatever sort on Monday, but allows it on the following day because it happens to be the birthday that tips him over the line that marks the allowed from the forbidden, seems to me to be indefensibly arbitrary. If someone can make the decision on the Tuesday, the chance that he was incapable the day before is small. The same applies to his incapacity on the Monday: quite why it should vanish magically on the stroke of midnight is a bit of a mystery.
I doubt one can set aside claims about permissibility so generally; it is utterly implausible to claim that one's reasons for thinking euthanasia permissible would have no ramifications for who can take that option, or the conditions under which they can take them. To take just the blatantly obvious example, one could very well hold that the permissibility transaction of this magnitude depends directly or indirectly on full powers of autonomous contract; there is no society that recognizes such powers in children, for a rather long list of reasons. And this is unsurprising; the reasons that establish something as permissible are what usually establish the conditions under which it is permissible. Thus Brassington's argument starts in the wrong place, and what is more, it can be seen that most of Brassington's argument depends crucially on doing so, because it requires the apparent symmetries created by simply proposing the permissibility of euthanasia as a hypothesis without any further grounds, so that this indefinite permissibility -- permissible under we know not what conditions -- gets treated as if it were general permissibility -- permissible under any conditions.
Even if we set this aside, the anti-ageism appeal is also problematic. As anyone can see from the argument Brassington gives, the direct implication Brassington's appeal is that there should be no age-based boundaries for anything. It's worth noting that this means that on Brassington's account the Belgian law is not legitimate. The law requires things in the case of children that it does not in the case of adults, simply because they are children. Brassington recognizes this, but seems not to grasp its implications:
That is, children still have to jump through administrative hoops that adults don’t. That may be defensible; it may even be morally required. Given the possibility that younger people are not as sophisticated as older, we perhaps would want to advert to someone who knows them well for reassurance that the request is authentic.
The general point, though, is that the law has been reformed so that it applies in more like the same way to adults and children. In many ways, that strikes me as a good thing.
But how could the anti-ageist appeal work here if it were even possible for it to be "morally required" for children to jump through administrative hoops that adults don't? How can "the possibility that younger people are not as sophisticated as older" even be functioning as a moral reason in the context of Brassington's argument? If it can be morally required for children to jump through administrative hoops that adults don't, it may well be the case that the administrative hoops that are required may well make it more practical to assume that people under a certain age cannot in general manage it on their own -- to go through administrative hoops presupposes and requires some designated competence for doing so, and nothing prevents one from judging that children will not usually be able to have such competence or, if they do, cannot be guaranteed to have it with sufficient ease to be taken into account under law. Likewise, sophistication is not a minor issue in matters of consent; if we are already allowing people to treat children differently based on sophistication when it comes to being able to take requests as "authentic", it becomes a question why we should not also consider the position that they simply may not be, in the main, sophisticated enough to make determining the authenticity of the request feasible.
In addition, we have to take into account the fact that anti-racism, anti-sexism, and in this case anti-ageism get their primary moral bite from being compensatory: they are intended to recognize inequalities leading to some populations being significantly more vulnerable than others. They also have the problem of being easily co-opted for any number of things that only sound like they might have to do with this kind of compensatory use. The real test of whether they are being used reasonably is whether they are being used explicitly and definitely in light of such vulnerabilities. But the kinds of things that Brassington's anti-ageist appeal would eliminate are precisely the things societies put into place in order to protect children as a vulnerable population. The reason we don't recognize children as having full competence in matters of contract is not that they can't make contracts but that others can easily take advantage of them. Children as inexperienced dependents can be pressured into doing things, or accepting things, that it would be much more difficult to pressure most adults into doing or accepting. (It is perhaps worth noting that Brassington's anti-ageist argument here is exactly parallel to certain common arguments against affirmative action as racist. And Brassington needs to address the same issue that people making those arguments need to address: how does one otherwise handle the vulnerabilities of the populations in question?)
A further problem with Brassington's argument is the conflation of legal and moral permissibility. The former depends in some ways on the latter, but they are in other ways rather different, and even the link between the two is not always straightforward. The fact that we pick a birthday as a cut-off point is purely a matter of positive law. It is not concerned with the question of actual competence or ability, because law is not about whether people are actually competent in some moral sense but about whether and in what practically feasible ways society as a whole can officially recognize that competence. In the case of children, for instance, the reason for age cut-offs is not because anyone thinks that suddenly everyone after age X is mature enough to drink, or drive, or what have you, nor that no one prior to this age is mature enough to do so; it's because these are thresholds that have been deemed to meet certain basic criteria, like:
(1) by that point we as a society have good reason to think that maturity has been reached by enough people that the law need not worry about the maturity issue as a general matter;
(2) the cut-off is relatively easy to determine by anyone who needs to determine it (as assessments of actual maturity rarely are);
(3) it allows enough time for parents and guardians to prepare people for the responsibilities, dangers, and risks involved in whatever kinds of decisions are in view.
Depending on the situation, there might be others. In other words, these age thresholds are never determined solely on the basis of moral permissibility. As legal thresholds they are deliberate creations through law of new distinctions between the permissible and impermissible, and they are based on a large number of factors. This makes quite clear another point at which Brassington's argument makes an illicit leap. Legal age thresholds are arbitrary in the sense that no considerations will establish exactly when the change-over point should be. For any age threshold, you could just as easily set it a month before or after whenever you do set it, for instance. But it does not follow from this that they are arbitrary in the sense that they have no underlying reasons. No age threshold is set on the assumption that a birthday magically changes someone; they are set on the basis of deliberation about a lot of practical assessments of the typical vulnerabilities of the population, of what's practically feasible for social recognition and for enforcement, and what will be least likely to mire society in nasty complications and consequences if anything goes wrong.
And, indeed, this is a problem with Brassington's argument in general. He says that the most important thing is capacity. But the capacity in question cannot be a mere ability to make decisions; by the very nature of the situation, it has be a capacity that it makes sense for society to recognize through law, given all the concerns society might have on the subject. That is, what matters is not whether the children in question are capable of making decisions but whether it is a good idea to extend a general recognition of such capability as a legal capability, given the ends of society, the vulnerabilities of children, and the practical limitations every society has to face. None of these are given any serious recognition by Brassington's argument, despite the fact that they are all involved in these questions.