Primarily he's arguing that we're trying to algorithmically model something that can't be modeled, and that the way forward is to look at the problem in a completely different way.

Is he not arguing that deterministic chaos is the underlying reason for this? The part I quoted literally has him talking about The Algorithm not existing a chaotic system. To quote wiki (emphasis mine):

Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems—a response popularly referred to as the butterfly effect—rendering long-term prediction of their behavior impossible in general. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as: "Chaos: When the present determines the future, but the approximate present does not approximately determine the future."

I think that's what the Three Body Problem is about:

One of the most famous is the three-body problem. Newton's theory of gravitation provides a simple solution to the problem of two mutually attracting bodies, for example the sun and one of its planets. However, as soon as a third body comes into play, for example another planet, the problem becomes mathematically unsolvable. [...] This is no longer true in the three-body problem. A tiny change in one of the variables, for example the speed of the planet Venus, might result in a totally different outcome, for example the planet Mars crashing into the sun. This is called "sensitive dependence on initial conditions".

From here. So I'd say Ben's argument is "the system is chaotic (=non-algorithmical) but deterministic (=computable) so there is no algorithm that can accurately predict the future, but we can probably predict the next minute by throwing truckloads of data at the problem."