Rebuilding interior stairs
If you see that the treads are what is giving and not the blocks supporting the treads, as isherwood says, there is no need for a center support, precisely for the reason he stated. Many stairs are built without center supports.

When I have my preference as to how I want to build stairs, this sketch illustrates my joining method.


All tongues and grooves are 3/8"X3/8" The groove under the tread since it is thicker 1 1/8", will leave plenty of mass to keep the nose intact. In your old tread, it looked as though the treads were glued up, but that glue joint failed, unless all are made that way... The idea is the groove will allow the treads to be better supported by the risers and when glued will prevent squeaking. The groove under the tread will allow a little give and take in the differences of the some of the risers, treads too, which will typically happen in hand made construction, if not by the differential shrinking of the support members.

If you have access to a table saw, I make the dadoes (grooves) by getting 4 or more blades the same diameter and stack them up on the saw arbor, if the saw has a cast iron arbor, until I get the width of cut I need. This is limited by the size of the cutout in the throat plate, but 3/8" should be attainable. Do not do this with a light weight portable saw.
Clean the original supports so the adhesive can be applied there to so nails will hold everything together until the glue set up. Stay off the stair system if possible until the glue sets go the glue tightens up all the joints. If the joints are moving before the glue sets, it will be a weak joint and may move again.