[6/8] feat(playground): wire up checkout mode

[tomiiide]

Which Linear task is linked to this PR?

Why was it implemented this way?

This is PR 6 of 8 in a stacked split of the original feat/widget-checkout branch (formerly the single ~16k-line draft #727). The work is broken into reviewable layers that merge bottom-up into feat/widget-checkout; each PR targets the branch below it.

Playground demo wiring for the checkout mode, folding the checkout toggle into the mode options. Playground packages are private (no changeset).

Visual showcase (Screenshots or Videos)

Checklist before requesting a review

• I have performed a self-review and testing of my code.
• This pull request is focused and addresses a single problem.
• If this PR modifies the Widget API or adds new features that require documentation, I have updated the documentation in the public-docs repository.

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Latest commit: 69cd6ce

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