It's incoherent to think the ability to reason requires the reasoner to be able to change permanently. You realize that LLMs do change; their context window and model weights change on every processed token. Not to mention the weights can be saved and persisted in a sense via LORAs.
The belief LLMs cannot reason maybe justifiable for other reasons, just not for reasons you've outlined.
I'm not sure you're right you know. I think that the way that an LLM maintains a conversation is to have the conversational thread fed into an instance of it at every step. You can see this if you do a conversation step by step and then take all of it (including the LLM responses) apart from the final outcome and paste that into a new thread:
if you think about how these things work as services you can see that this makes sense. The model weights are several gb, so caching the model weights for utilisation by a particular customer is impractical. So if the forward pass does update the model then that's instantly discarded, what's retained is the conversational text, and that's the bit that's uploaded to the model on each iteration for a new reply. There are hundreds of requests pinging through the data center where the models are used every second, all of these use the same models.
But if you believe that there is a reasoning process taking place in the text then fair enough.
The belief LLMs cannot reason maybe justifiable for other reasons, just not for reasons you've outlined.