Draft: Update Pajamas docs live code examples to more realistically match usage

What does this MR do?

Reworks the live code examples across 32 component docs pages so that what you copy out of the docs looks like expected product usage instead of a minimal demo: realistic data, state where the component actually needs it, and more of the prop combinations people implement in the product.

This is the WIP I said I'd share on the 2026-09-01 Design Expression team call. I'm opening it as a draft so we can start collaborating and evaluating; it still needs a cleanup pass and may be worth splitting before review.

Part of #3575.

Split into a stack of 14 MRs

Now that everyone's happy with the content, I've split this draft into a stack of 14 sequential MRs, each one safe to publish without the ones after it. The combined stack reproduces this MR's diff exactly. I'm keeping this MR open for reference while the stack merges, then I'll close it without merging. No further commits will be pushed here, so review the stack instead:

1. Docs examples 01/14: Update live example infras... (!6312 - merged) • Chad Lavimoniere • 19.5 (live example infrastructure)
2. Docs examples 02/14: Rework button and button g... (!6313 - merged) • Chad Lavimoniere • 19.5 (button, button group)
3. Docs examples 03/14: Rework dropdown combobox e... (!6314 - merged) • Chad Lavimoniere • 19.5 (dropdown combobox)
4. Docs examples 04/14: Rework dropdown disclosure... (!6315 - merged) • Chad Lavimoniere • 19.5 (dropdown disclosure)
5. Docs examples 05/14: Rework checkbox and radio ... (!6316 - merged) • Chad Lavimoniere • 19.5 (checkbox, radio button)
6. Docs examples 06/14: Rework text input and form... (!6317 - merged) • Chad Lavimoniere • 19.5 (text input, textarea, select, form group, form input group)
7. Docs examples 07/14: Rework form structure and ... (!6318 - merged) • Chad Lavimoniere • 19.5 (multi-step form, form fields, date picker, password input)
8. Docs examples 08/14: Rework toggle, search, and... (!6319 - merged) • Chad Lavimoniere • 19.5 (toggle, search, sorting, segmented control)
9. Docs examples 09/14: Rework modal and drawer ex... (!6320 - merged) • Chad Lavimoniere • 19.5 (modal, drawer)
10. Docs examples 10/14: Rework messaging component... (!6321 - merged) • Chad Lavimoniere • 19.5 (popover, tooltip, alert, toast, banner, broadcast message)
11. Docs examples 11/14: Rework navigation componen... (!6322 - merged) • Chad Lavimoniere • 19.5 (breadcrumb, tabs, pagination, path, nav item, link, collapse, accordion)
12. Docs examples 12/14: Rework table, card, and da... (!6323 - merged) • Chad Lavimoniere • 19.5 (table, card, dashboard panel)
13. Docs examples 13/14: Rework avatar, badge, labe... (!6324 - merged) • Chad Lavimoniere • 19.5 (avatar, avatar group, badge, label, token, skeleton loader)
14. Docs examples 14/14: Rework progress and utilit... (!6325 - merged) • Chad Lavimoniere • 19.5 (progress bar, spinner, infinite scroll, utilities)