[Story] Public rooms search

[Velin92]

Story

This story is related to the first deliverable for BWI for Apr 2024.

NOTE: The public room search won't be publicly available, but will be an internal only feature.

Requirements

• As a user I want to be able to look for all the available public rooms on my homeserver
• As a user I want to be able to filter/search the list of available public rooms on my homeserver

Leads

Tech: @Velin92 Design: @americanrefugee

Size Estimate

M

Designs

https://www.figma.com/file/TKbfHw1zboMfAfP9xxSTsF/Search?type=design&node-id=249-61389&mode=design&t=Y4AdaYGpeDyILBZa-4

Out of scope

• Searching for rooms that belong on a different homeserver other than the user's own.
• Sorting and filtering are not required
• Also since we are only searching on the user's own homeserver, organisation categorisation is not needed

Open questions

### Questions
- [x] ~~Should we use a dynamic filtering like in the room list, or should we have the server make the filtering like the public room search of EL-Web?~~ We need pagination since public rooms can be 100 of thousands
- [x] ~~Designs? Without those is hard to estimate iOS and Android work, and create tasks for them~~ We have some designs however we should keep in mind that for this their scope is reduced (check out of scope)

Subtasks

### Rust
 - [ ] https://github.com/matrix-org/matrix-rust-sdk/issues/3154

Subtasks

### iOS
 - [ ] https://github.com/element-hq/element-x-ios/issues/2518

Subtasks

### Android
 - [ ] https://github.com/element-hq/element-x-android/issues/2474

[Velin92]

Cloned from: https://github.com/element-hq/element-internal/issues/559