feat(mosaic): default Profile.NavItem badges to neutral via BadgeContext

[alexcarpenter]

Description

Stacked on #9856.

Adds a BadgeContext to Mosaic so a parent can set a default color for the Badges rendered inside it, following the ButtonContext pattern. An explicit color on the Badge still wins.

Profile.NavItem provides color: 'neutral', so a nav item badge can be written as:

<Profile.NavItem value='security' badge={<Badge>2</Badge>}>

The Profile swingset docs and stories drop the now-redundant color='neutral'.

Checklist

• pnpm test runs as expected.
• pnpm build runs as expected.
• (If applicable) JSDoc comments have been added or updated for any package exports
• (If applicable) Documentation has been updated

Type of change

• 🐛 Bug fix
• 🌟 New feature
• 🔨 Breaking change
• 📖 Refactoring / dependency upgrade / documentation
• other:

[vercel]

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Project	Deployment	Actions	Updated
clerk-js-sandbox	Ready	Preview	Sep 22, 2026 2:41pm UTC
swingset	Ready	Preview	Sep 22, 2026 2:41pm UTC

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🦋 Changeset detected

Latest commit: a6a21ee

The changes in this PR will be included in the next version bump.

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[prestonwebdev]

Looks good to me!