Your assessment is correct. No flush valve.
The flush can be done that way, but why? A valve is literally pennies at the bulk prices they are probably paying. Worst case say it's an extra buck (or £), it's a worthwhile investment for ease of use. Who wants to remove the membrane everytime they swap carbon blocks.
Good news it's an easy adjustment for the future. Double good news the flush valve is not your current issue with the unit!