Given your understood need for local-only AI, I strongly recommend watching the video for the 6 Jun 25 meet-up where physician Dr Andy Bland described his local-only set-up and the toolchain involved. In his case patient record confidentiality (HIPAA) was a concern/constraint.
The MCP connection aspect of AI work is very new. My understanding is that the MCP approach (i.e. AI+app integration) debuted in the Anthropic (Claude) community so using Claude as the initial test context for Tinderbox makes sense (by way of managing expectation). A lack of a local/private implementation for Tinderbox+AI is just state of the art ATM not a policy choice.