I've never seen tread brackets used on the interior skirting of stairs, but I linked this picture of the tread brackets in case it is useful. Has anyonne seen tread brackets used where peoria is talking about? it would be handy for covering up those gaps, but I just can't wrap my head around using it there.

If you use stainable filler, tape the edges of any adjacent wood you want to stain, because any smudges of filler that get onto the wood will show, even after sanding, and it takes the stain differently than the surrounding wood. Also gel stain will blend the filler into the surrounding wood better than most other stains. It's a pain.

Here is a link that might be useful: tread brackets

Peoria, I'm not quite picturing where your problem is--Slateberry's answer is totally different from what I was thinking. Could you clear things up?

You say the treads are housed--to me, that means that they fit into the stringer and trim board--your balusters do not rest on the tread, but on the side trim of the stairs in a straight diagonal line, like at the top of the railing.
I was picturing the gap as being visible from the top of the tread--that it didn't fit into the trim all the way from front to back--and I couldn't see how the front could fit on every step, but not the back of each?

If the treads are open (visible) on one side, with the balusters resting in pairs on each step, then the gap is zero at the front, but wider as you approach the front of the next step, so the gap runs horizontally? It would seem to me that if every step is that way, then they were either poorly done to begin with, or the outside trim was replaced at one time, and the cuts were off.