How do you write a web tool that lets users configure and combine arbitrary third-party APIs, including those not known or not even existing at the time of development, into a custom solution that runs in their browser?

Answer: you don't. You can't, you shouldn't, it's explicitly not supported, no third-party API provider wants you to do it, and browsers are designed to actively prevent you from doing such a thing.

That's the core problem: MCP has user-centric design, and enables features that are fundamentally challenging to provide[0] with a network of third-party, mutually mistrusting services. The Web's answer was to disallow it entirely, opting instead for an approach where vendors negotiate specific integrations on the back-channel, and present them to users from a single point of responsibility they fully control.

Doing the same with MCP will nerf it to near-uselesness, or introduce the same problem with AI we have today with mobile marketplaces - small number of titans gate-keeping access and controlling what's allowed.

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[0] - I'd say impossible, but let's leave room for hope - maybe someone will figure out a way.