EDT automatically assigns students to specialization groups. Can students still be assigned manually?

This is possible, provided you delete the links generated automatically between class parts if you have let EDT generate the groups, as these links become blocking when you assign them manually: in the display Timetable > Classes > List, launch the command from the menu Edit > Delete the optional links.

Then add students to the display Timetable > Groups > Students as for any group: select a group on the left, and right-click on the line Add a student.