If by local MCP you mean stdio, then a tailnet won't help you unless you proxy the local MCP using something like mcp-proxy: https://github.com/sparfenyuk/mcp-proxy . That local mcp is only exposed to apps on the machine. The flow is Phone client --> agent harness or whatev --> Tailnet --> Mac Mini --> mcp-proxy http listener --> local mcp. But you need to wrap the local mcp runner in the proxy (check the docs).

EDIT you don't mean running on your actual phone lol, you mean in the web browser at Claude.ai. Anyway, if you use mcp-proxy for a stdio mcp, or if it's an http mcp alone, then you would need a public endpoint for Claude.ai to connect to. Tailscale will only help you if you use Funnel, but this cloudflare thing is exactly what you need (w/o Tailscale).

I built something for this exact use-case using Cloudflare Tunnels:

https://github.com/dalberto/mcp-ferry

I also use CF managed auth to make auth easier.

if both your mac mini and your phone are on the same tailnet, can't you just ssh to your mac mini from your phone's terminal emulator and have claude code/codex run on it?