The way that foundations are built today is, you dig a hole about five feet bigger than the house, when it is back filled it is seldom compacted and often sinks over time.
As a way to combat that people tie sidewalks and poarches to the foundation.
I think that had more to do with this problem than your downspout. That's just my opinion any way. With your first patch job leaving things in place, there continues to be stress on the wall and sooner or later you will have another crack toward the other
end of the poarch. Mud jacking the poarch should push the wall back into place but the crap that has found itself inside the crack will make that a problem.
If there is rebar in the wall, it has now been damaged with water and air and will have started to rust causing more damage to the concrete near the crack.
So I would go with the guy that wants to remove some on both sides. I would add rebar to tie new to old but I would also require that after it has been smashed out I would use a diamond blade to make a nice clean cut on the outside so a gasket can be added. I have only used it one time , when poring a foundation in two peices we stuck this stuff to the face of the first one to seal it agains water. Black foam, peel and stick 1/4"x 1".
When mudjacking the poach, do spots for the whole twenty feet to make sure this isn't an ongoing problem.