Note that Claude Desktop can be run on Linux using https://github.com/aaddrick/claude-desktop-debian, which repackages the Windows version to run on Linux (it's an Electron app, so it just needs to stub out the native interface, which seems mostly for cosmetic things).
It would be really nice though if the web versions of Claude, ChatGPT, etc. added MCP support directly: this should be achievable with help from a WebExtension and a native binary to proxy messages from the WebExtension to the MCP server.
It should also be possible to write such a WebExtension by a third-party (that injects messages in the conversation), although it's possible the LLM companies might block it.