Do you consider yourself a successful AGI? I guess you do.
Does that make you always right? No.
Does that make you at least capable of staying rational in all situations? No.
Well maybe the logical framework that you (and I) are basing all of rationality on is at least coherent? What do you mean it is not[1]? Who's that Gödel guy anyway?
The issue at the heart of this definition of AGI is that it is undecidable. That is why rather than proving that an intelligence is unambiguously succesful at being general, tests are being used, which look at whether it succeeds "often enough" at a given task in a particular set of conditions. The Turing test is such test, but it is not the first one ever devised, the Jewish golem was not able to talk and did not get a name, so it was considered "unfinished" and as such failing the "being human-like" test. I don't want to bring unrelated parallels to this post, but the key takeaway I think is this one: it is ultimately fruitless to define intelligence in terms of "success", as if it was a physical quantity one could measure.
So now to get back to your original question, chat AI have had large amount of success in the past year, one Google researcher who worked with them is even assuming some of them are sentient [2] and makes some strong arguments in that direction. AI can drive cars, play videogames, hold conversations... it's less than what humans can do, but they can also fold proteins[3] and find new patterns in humongus amounts of data [4], something that was impossible to do in a reasonnable amount of time before they were created. So are they succesful? I would say yes; are they perfect? Oh no far from it, and my point is that they will never be perfect, and will always fail in some regards.
[1] https://en.m.wikipedia.org/wiki/G%C3%B6del%27s_incompletenes.... [2] https://www.scientificamerican.com/article/google-engineer-c... [3] https://www.science.org/doi/10.1126/science.370.6521.1144 [4] https://theconversation.com/seti-alien-hunters-get-a-boost-a...