No, you are not missing anything.
First, the Custom Check names should change in the Title Information window after you change it. I will mention this to HC and see if this can be fixed.
As for using Title information in Advanced Find, there currently is no way to select Title fileds from a drop-down box. There is only Issue fields.
However, as an unsupported function, you can use Title fields.
(I have updated the Advanced Find manual in 'News & General Discussion' to reflect this)
You do this by using the Table name instead of the I (which is for Issue Table).
For example, using an Issue field it would look like:
I.Notes
For a Title field use either ComicTitles, BookTitles, or MagazineTitles, depending on the Type you are searching, like this:
ComicTitles.Publisher = 'Marvel'
BookTitles.Publisher = 'Ballantine'
MagazineTitles.Publisher = 'Time'
ComicTitles.[CustomCheck1] IS TRUE
BookTitles.[CustomCheck1] IS TRUE
MagazineTitles.[CustomCheck1] IS TRUE
This is not supported by HC and I haven't tested it on all Title fields.
As for Reports, you can't use Title fields if you used menu item Reports->[type_of_report] but you can run a report on the result of a Find.
Run the Advanced Find to get a result.
Select some or all of the issues.
Right-Click and from the pop-up menu, select either 'Quick Report' or 'Report Selected Titles As...' then either 'Collection Report', 'Item Checklists', or 'Price List'. (I couldn't get the 'Collection Overview' or 'Wanted List' to work.)