I would claim that, if you think without introspection (that loop), then there is virtually no self check. I'm not sure what "self check" you see that the brain has. Could you describe this "self check in a line of thought"? How do you perceive the check there? This is a genuine question. It definitely doesn't align with how I think about things. I ponder and talk to myself to iterate verify and test my understanding of my own thoughts.
Maybe a good analogy is "throwing a paper plane in real life" and "throwing a paper plane in a video game".
In real life, the paper moves "by itself". It does not need an external loop that update its position in a loop manner.
In the video game, you need an internal loop, a step-by-step tick, that update the plane position based on its current position and its momentum. And this is why a video game paper plane is not a real object. It is a very good simulation, it looks like it, but it is missing some intrinsic properties that we expect from a real object.
Yet you can analyse the paper plane trajectory and see it as a Markov chain, with quantified step-by-step progress (for example one position point every 0.1 second). The same way you can look at your though process and identify a step-by-step progression. But it does not mean that it works like that intrinsically, it does not mean that the paper plane "jumps" from position point at time T1 to position point at time T1+0.1 second.
For the human brain, there is no "loop centre" in the brain. There is no one (to my knowledge) who got a brain injury and suddenly were unable to keep a single line of thought without having someone else having to feed them the previous thought in order to feed the next thought.
In the brain, the fact that the previous thought feeds the next thought is "how it works", it is intrinsic, it is by design. And this mechanism of thoughts feeding the next thoughts is what creates "consciousness" or "awareness": self-reflection is based on the fact that thoughts are intrinsically linked together, that they "flow" continuously, without needing an external system to update them.
You cannot take away the "loop" part of the paper plane so that it suddenly would be unable to move on its own once thrown away.
Now, you can always say "well, the paper plane in the video game is a very good simulation, it does not matter if it is a real object or not", and that is fair enough. But in this discussion, some people have arguments to support that this property matters, that it is one condition for consciousness or awareness.