Can we circumvent the private access rules without relying on techniques that are considered “arcane” and likely to be made illegal (at some point)?

I recently read the article Profiting from the Folly of Others by Alastair Harrison, which walks through a technique to circumvent private access rules, particularly by means of (ab)using the template instantiation mechanism together with injections of inline friend definitions. I first saw this technique used in Filip Roséen’s blog article series on stateful metaprogramming back in 2015, which made use of it to implement a constexpr-counter that were, if not fully, at the very least nearly¹ standard compliant. A. Harrison’s article mentions that the origin of the technique is likely two blog articles by Johannes Schaub published in 2010-2011².

What is an example of a non-deduced context, and why can it be useful to know about these?

The following snippet

template<typename T>
struct Foo { T t; };

template<typename T>
void addToFoo(Foo<T>& foo, T val) { foo.t += val; }

int main() {
    Foo<long> f{42};
    addToFoo(f, 13); // error: no matching function for call to 'addToFoo'
    return 0;
}

is ill-formed, as (function) template argument deduction for the dependent function parameters foo and val of the addToFoo function template resolves to different, conflicting types in the addToFoo(f, 13) function call.