In case anyone is interested, I got a reply from Chessbase who confirmed that they have indeed changed things - to accommodate mobile users. However, the good news is that the more functional version is still available as well if you know where to go
This is what they said:
"Our developers had to customize the functions for the use on mobile devices.
The old version is still available at https://database2.chessbase.com
Sort for moves should be possible also in the new version.
Position search will be integrated later."
A few days ago online Chessbase Database started to be presented in a new layout when I logged in. Do other people see the same or I have been diverted to some sort of beta test?
Importantly, this new format does not allow you to put in a position and simply click "Search Board" to see the games that reached this position and how they continued.
(For reference, the one I am seeing now has a grey bar at the very top with "DATABASE by ChessBase" in the middle and underneath a place for players' names and photos.)