The Check Your Games feature is not a replacement for position search, rather it's a completely new feature. Position search is now integrated into the database page. For Chess Publishing Updates, that is
https://www.chesspublishing.com/database/chesspub. Similarly, for TWIC it is
https://www.chesspublishing.com/database/twic. Links to the different databases are available from the dropdown Databases menu in the main navigation bar. On your first visit to the database page, you should be presented with a quick walkthrough of the features that can be reopened at any time via the Help icon (?).
There is a list of games bottom left of the database page, that can be filtered. The Position Search "Auto" toggle, when turned on, will update the list of games automatically as the position on the board changes. Or you can hit the "Search position" button in a specific position and keep the Auto toggle off to "pin" the search to a specific position.
Apart from the list of games, there is also a list of Updates that updates automatically with the position on the board. And the opening moves explorer is to the right of the screen.
On mobile devices, the layout is different, but the same functionality is available.