Your example is also interesting, but doesn't really break things in the same way.
I don't necessarily agree with this, I think there are plenty of important things to consider with both, but that's why one became famous without the other.
If we can follow sound reasoning such that some statement can be either true or false, based on the exact steps we take, that is exactly the same problem: a contradiction somewhere.
"This sentence is true" doesn't contradict itself, but two different ways of evaluating it contradict each other.
Researcher A assumes the sentence is true and comes to the conclusion that it is.
Researcher B assumes that it is false, substitutes in the value and also concludes that it is correct.
Neither has found a contradiction in isolation, but A and B's findings contradict each other.
In terms of electronics-based intuition, "this sentence is false" is like an unstable circuit: e.g. an inverter feeding its own output to its output. This might result in oscillation. "This sentence is true" is like bistable feedback: it gives us a flip-flop which "latches" whichever value we inject.
It's consistent; but the problem is that it's that way whether we assume it to be true or false, oops.
For some reason, "this sentence is true" isn't famous, like its cousing, the Liar Paradox, even though it has something significant to add.