I believe I understand everything you are saying and why you are saying it. I think you are completely missing the point, though. LLMs already do quite a few things they were not designed to do. Also, your idea of sentience seems very limited. Yes, with our biological computers we have some degree of presence over "time", but is that critical - or is it just critical for us due to our limited faculties.
What if "the internet" developed some form of self-awareness - would we know? Our entire society could be subtly manipulated through carefully placed latency spikes, for example. I'm not saying this is happening, just that I think you are incredibly overconfident because you have an understanding of LLMs current lack of state etc.
If we added a direct feedback mechanism - realtime or otherwise - we could start seeing more compelling emergent properties develop. What about feedback and ability to self-modify?
These systems are processing information on a level we cannot even pretend to comprehend. How can you be so certain that a single training refinement couldn't result in some sort of spark - curiosity, desire to be introspective, whatever.
Perhaps Hofstadter is losing his mind - but I think we should at least consider the possibility that his concern is warranted. We are not special.