You absolutely can make a local MCP server! I use one as part of TalkiTo which runs one in the background and connects it to Claude Code at runtime so it looks like this:

talkito: http://127.0.0.1:8000/sse (SSE)

https://github.com/robdmac/talkito/blob/main/talkito/mcp.py

Admittedly that's not as straight forward as one might hope.

Also regarding this point "letting you easily connect to your own Agents SDK MCP servers deployed in their cloud" I hear roocode has a cool new remote connect to your local machine so you can interact with roocode on your desktop from any browser.