Wildly understating this part.
Even the best local models (ones you run on beefy 128GB+ RAM machines) get nowhere close to the sheer intelligence of Claude/Gemini/Codex. At worst these models will move you backwards and just increase the amount of work Claude has to do when your limits reset.
That said, Claude Code is designed to work with Anthropic's models. Agents have a buttload of custom work going on in the background to massage specific models to do things well.