Short answer: not in the way you’re describing, at least not today.

What you’re comparing are really two different product shapes:

• Codex = a more opinionated product/runtime with its own supported model path and UX
• Continue / similar tools = a model-routing shell where the UI is built to let you swap providers/models more freely, including local ones

So if your goal is:

• “keep the Codex UI”
• but “plug in any remote or local LLM I want”

that is generally not the current model of Codex.

A more accurate way to think about it is:

• if you want maximum model flexibility, tools like Continue are closer to that design
• if you want the Codex-specific experience/workflow, you usually accept the tighter model/runtime constraints that come with it

So I would not expect Codex to behave like a generic bring-your-own-model frontend unless the team explicitly adds that as a supported capability.

If your real goal is local/private inference with a nicer UX, the practical answer today is probably:

• use a tool designed for multi-provider/local-model routing,
• or keep Codex for Codex-native workflows and use a separate local-model tool for the BYOM use case.