Integrate searching public rooms and people into the new search experience

[kittykat]

To provide a seamless experience, "Explore Rooms"/"Public Rooms" and DMing one person should be part of the new search experience.

Integrating or making any changes to the "DM"/"People" dialog is out of scope.

Designs

• Design issue
• Figma

Tasks

• Implement the designs
• Test with the team

[t3chguy]

Closing in favour of #21352

Nvm this is the development task

Design task: #21352