It seems there are some relatively short edges near the top of the cross shape. I think the line tools is preferring to snap out of plan. Turn on hidden and zoom in a bit. I was able to get the top face to reform after two lines.
I also intersected the geometry of the cross part first.
select both cylinders, entirely, and ‘Reverse Faces’.
‘Select All’, and select ‘Intersect Faces / With Selection’
Select tops of cylinders and delete.
Select the 4 intersecting lines that cross at the axis.
Select entire surface.
Now, apparently to circumvent a bug in the software and get the software to start thinking correctly again, draw a line over one of existing lines?
At this point the crossing groove surfaces should be selectable without also selecting the entire surface.
delete top cylinder surface, and you should be left with only the crossing grooves with connecting remnants which you should carefully delete.
Wow. It’s a bit weird, and you’ll find that if you get just a bit more complex than this, like drilling intersecting holes through the plate, SU really gets messed up.