I’d leave the public-facing titles (the_title()) alone so you don’t have to mess with filtering things if you change plugins or themes. Adding postmeta is not a big deal, and it’s not hard to maintain/sustain. It also will not get removed/overwritten with updates. The biggest benefit here is that if you forget to add a custom title, it will only affect you. If instead you try to use something other than the_title() in the public-facing version, if you forget to add the custom title, you could end up accidentally displaying a post with no title.
On the editor screen, you would add a custom postmeta box. Then in the editor view, you would type in your “admin title” in this meta box. Alternatively, you could choose to use a custom fields plugin to make this easier to accomplish – some of these plugins will allow you to mark a field required, so that if you forget to enter the “admin title” when you publish a post, you’ll get an error message reminding you to fill it in before it will let you publish.
From there, you can add a new “admin title” column in wp-admin – this is called a custom list table column. You can even make it sortable to alphabetize by your “admin title” rather than by the_title().