I have noticed this as well. Its very frustrating as its a core part of mcp apps. I had to code around this limitation essentially killing some features of my app.
I built a test app where the only action was to test to see if a secret string could be read out of chat through update model context and it was not working.
One thing to note is that I found if I refresh the page everything starts working. My theory is that there is some sort of race in the initialization code on the ChatGpts side of the message bridge.