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This article overstates the problem quite a bit by saying, for example, "Only non-Turing-complete languages support formal reasoning and verification".
The point of Turing's result and related theorems is not that you can't reason formally about programs, or understand or predict what they do, or prove that they are correct. It's that no automated method is powerful enough to decide nontrivial properties for every program; there are always programs for which the decision procedure will either say it doesn't know, or be wrong (or the decision procedure will take an infinite amount of time).
However, there are automated methods that can decide nontrivial properties for many programs, and the existence of programs where a given property can't be decided doesn't mean that the answers, when they exist, have to be wrong.
We do have specific programs in Turing-complete languages whose behavior or whose correctness to a specification is proven, and formal methods that can be applicable to them.
I think the article's conclusion might still be right, though: having environments where you can't always determine correctness may be playing with fire, so it may be a better choice to avoid that kind of risk entirely. But that doesn't mean that, given a program, we're always going to be completely in the dark about what the program does!
A thing that you could do for smart contracts is have formal-methods analyzers that check properties that you care about in the contract. The analyzer can say, of a particular contract, "good", "bad", or "don't know". Then you can avoid (or even somehow ban?) contracts that get "bad" or "don't know".
Or, you could simplify things by having "bad" and "don't know" be in the same bucket! You could call it "can't be proven safe". Because of the halting problem issue, some safe contracts will always end up "can't be proven safe", but if the analyzer is good enough to allow for quite a lot of flexibility in the contracts, you can say that writing something that can't be proven safe is the contract author's fault or responsibility.
Then you can say that you will only use contracts that can be proven safe. There's no contradiction between this and Turing completeness or the halting problem, and there's no inherent reason to think that these contracts will be rare or hard to find. You do have to be careful about what particular properties you're checking for in your definition of "safety", though!