Yep, you will need something to funnel the water into the culvert and keep it from swirling and eroding the banks. You can see they did this with the concrete in the second pic you posted. What probably killed the original installation though was the use of 2 culverts instead of one. Water has great mass, and the water flowing down the middle of the creek is held there by gravity and the water flowing along on either side. When it hits those two culverts, the water that is in the middle is forced into the triangular opening below the center formed by putting 2 round pipes together. Since it is moving it has stored energy and flows into and builds pressure in that space. This flushes out the material there, kind of like stuffing a garden hose into a gopher hole...
If you do this again, use one larger culvert with inlet wingwalls/funnels, or if using two, space the culverts a little ways apart and do something similar with concrete between the culverts. If this is not possible, then a plate should be added that joins the two circular passages together all the way to the creekbottom, with a sill plate at the bottom that extends a ways upstream and will keep water from flowing between the pipes or under/behind that plate.
This link is for box culverts, but the wingwall diagrams apply the same for round culverts.
New Focus for Highway Safety, September/October 2005 Public Roads