Could we include a search bar to pick any SE site to migrate to instead of just 2 options?
This is actually what we mods have. While regular users have only a limited list of options, moderators can migrate a question to any site on the network they choose - even other sites' Metas! Of course, since my close-votes are binding, I have to be absolutely certain I'm sending a question to the right place when I do this.
My understanding is that the migration paths for regular users are only for the most common migrations, so they can be handled by the community without the need for a mod to step in. Mods are then only required for the (theoretically) rare occasions where a question needs to be migrated to a site that's not on the list. The list can be extended, but only to a maximum of five or so.
My suggestions to you would be:
- If you want a question to be migrated to a site that doesn't have a migration path, flag it for moderator attention with an explanation of which site you think it should be migrated to, and we can take care of the rest.
- If you want a new migration path to be added, you can start a separate discussion here on Meta and state your case. (I'd personally advocate a migration path to Literature.SE, but would like to check the stats for how many questions we've migrated there.)
- If you want the migration search function for moderators to be made available to regular users, you would have to post a feature request on Meta Stack Exchange. That would likely be a network-wide change, and isn't something I have the power to do.