We installed Hardiebacker board - we screwed it every 6-8 inches and did not use a thinset to adhere it.
There's your first mistake. Your tile can crack when you don't follow the instructions and do what you did.

I can understand why we would fill the joints but not why we would tape them as well.
You tape the joints because that is also the wishes of the manufacturer. You tape the joints to reinforce the integrity of the strata. No tape on the joints means cracks in the tiles over the joints.
Can the filling between the Hardiebacker panels be done at the same time as we are installing the tiles?
Normally YES. But not before you remove the backer and start over installing it in fresh thinset.

all of the edges of the joints between the Hardiebacker panels are level
Doesn't really matter at this point.
what is the purpose of the tape if there is no 'feathering' required?
See above explanation.
