I don't really get constructor theory, but if this is a crank website it has to be the one with the best pedigree.

David Deutsch has done a lot of impressive work, like coming up with the first universal quantum algorithm and providing a model that proves quantum mechanics is local.

I'm not basing my judgement on the pedigree, I'm basing it on the content. What constructor theory is about is assigning and re-assigning concepts and language to existing physical theories. That's called "metaphysics," a practice that believe it or not predates physics. (Philosophers have been formally assigning concepts and language to informal physical theories long before the first formal physical theory was articulated.) It's not like Deutsch is in bad company, Aristotle also wrote about metaphysics.

Nonetheless, that does nothing to change the fact that crank websites almost always share three elements: they are works of metaphysics but claim to be physics, they have way more popular writing than academic writing (in this case, see the size of the media page vs the empty publications page), and they have a website advertising them (that's actually pretty unique to cranks, for some reason). I don't really have an axe to grind, it's just that the linked page is a typical crank website.

why? You don’t explain.

I'd agree that at first glance this reads like crank material and the question to answer is "why not?" instead of "why?". It is extremely vague and doesn't parse into anything that seems meaningful.

If something seems that way to you, and you want to say to others that it seems that way, you could explain why.

It’s not like it was developed by random nobodies, and it’s not like the default assumption should be that some work is the result of cranks (the default assumption should be agnosticism towards it)

https://en.wikipedia.org/wiki/Constructor_theory

https://en.wikipedia.org/wiki/David_Deutsch

The implication is that almost any practicing physicist would parse this as crank-ish, so we can discuss it operating from the shared position that it needs to be defended to be worth spending time on. Similar to how most physicists are not going to spend any time responding to random crank emails, because they parse as cranks; the burden of credibility is on the writer.

Credentials offer some credibility, but imo not enough to make this seem worth spending time on.

Anyway, if I had to put the criticism into words: the entire page is weird jargon that doesn't appear to solve any problems that we have, puts a simple problem into inscrutable metaphysical terminology, and the authors didn't feel like they needed to provide any examples of their model being useful or providing any insights. So my reaction is to ignore it until they do the work of showing why we should care.