Simple search any is the best approach. It uses LIKE with wilcards in the database query allowing for partial searches, but it's not going to fix someones typo.

Auto complete for text input search usage is something we'll likely add later when we've refactored our field classes so things are structured better to allow us to more easily add new features to field behavior. For now it'd be a massive amount of duplicate code and headache to implement as it'd have to be done on a field type by field type basis.

I've added our internal plans for the field rewrite as a ticket on forge to track the rewrite of field classes, but it won't be for awhile most likely as you'll see it's a ton of work to do.

forge.joomlapolis.com/issues/5785

I've also added a feature ticket specifically for auto-complete for text types.

forge.joomlapolis.com/issues/5786