Rework admin organization selector

Possible features:

- [ ] Modal for organization picker
- [ ] Show name, acronym, verified status
- [ ] Filter verified/unverified
- [ ] Searchable by name and acronym
- [ ] Show all orgs in list
- [ ] Have a `recent` area for 5-10 most recent orgs. Can store as something like `adminOrgRecents` on the user document
- [ ] Remove org from `recent` if org deleted
- [ ] Remove org from `recent` if affiliation changed below admin
- [ ] Remove org from `recent` if affiliation removed
- [ ] Create new mutation like `ADD_RECENT_ADMIN_ORG` to send when navigating to org information page
- [ ] Add button to remove org from `favourites` with a click (no verification required)
- [ ] Include `accessedAt`
- [ ] Include toast message for action
- [ ] Optional: Add `undo` to toast. Just recall `ADD_RECENT_ADMIN_ORG` again with the required details
- [ ] Have a `favourites` area. Can store as something like `adminOrgFavourites` on the user document
- [ ] Remove org from `favourites` if org deleted
- [ ] Remove org from `favourites` if affiliation changed below admin
- [ ] Remove org from `favourites` if affiliation removed
- [ ] Include `favouritedAt`
- [ ] Include toast message for action
- [ ] Optional: Add `undo` to toast

No files, tests, or entry points are named. Start by locating the existing admin organization selector, organization data, user document, and mutations; then map the proposed modal, search and filtering, recent organizations, favourites, lifecycle removals, and toast actions. Done means the agreed subset is implemented and its behavior is covered by the project’s relevant tests.

Written by the indexing model from the issue text.