Here's what worked for me:

I was only able to run the Inspector correctly in VS Code's Simple Browser (Ctrl + Shift + P → Simple Browser: Show → Paste your Inspector link—in my case, it was running on http://127.0.0.1:6274).

I still have no idea what the problem is. I couldn't make the Inspector work in Edge or Chrome. It seems I have tried EVERYTHING: replacing STDIO with SSE, running the MCP server with SSE transport on Ubuntu server and connecting to it from the Inspector running locally, cleaning cache, opening the Inspector in Incognito mode, stopping other processes even though I couldn't find anything messing up with the ports needed, etc., etc. Nothing helped but simply running the Inspector in VS Code Browser did. It works !! I am really confused how there aren't any discussions on this. Hope there's something that might need to be fixed on the SDK/Inspector's side