Search should return private rooms where the bot is already in or invited to (`world_readable`)

Nobody has claimed this yet.

I made a private room where I invited the bot (or even have the bot joined in from the past) I expect that room search on the archive site should return this room as it is able to show its chat log.

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_See below for some background information._

_If I invite the bot to a room that is invite-only but has history visibility set to public, it would be desirable to show the chat log via the web._

_You may ask why this odd combination is beneficial. It's a new proposal against scammers for public groups. Anything else other than invite-only allows for spamming and flooding room state, and we proposed creating a CAPTCHA bot that would let you speak this way, while still retaining readability of the room for outsiders._

https://archive.matrix.org/r/mod-ideas:matrix.org/date/2023/06/22?at=$Y42s_vlxoOy_JRm71sKx09a8gguAgK-fKjRe29FtQXk

Start by tracing the archive site's room-search entry point and the filtering logic for world-readable rooms, then inspect how the viewer determines whether the bot is joined to or invited into a private room. Done means search results include eligible invite-only rooms with public history when the bot is already present or invited, while existing room visibility rules remain intact.

Written by the indexing model from the issue text.