Perhaps some members have successfully patched concrete to a durable long lasting surface/product. In 35 years of commercial construction I have seen zero successful “fixes“ that lasted any length of time and did not look like a patch job……and I don‘t experience the freeze/thaw/cycles and salt that you likely do. There are more concrete patch products and promises of long lasting fixes than oil and fuel additives

. I speak only from experience, not in absolutes, so hopefully any members with success stories will chime in.
With that out of the way

if you want a long lasting solution for the concrete you will need to completely remove a section, drill and install dowels to the existing (which may not be thick enough to successfully accept this) and pour back - I’m thinking 6” thick or so.
In another thread on this subject a member bought a tube like seal for the bottom of the door and installed a garden hose in it to help it keep its shape.
I would be interested in looking at cutting off part of the door bottom and scribe a piece of pressure treated lumber to the bottom. Other suggestions have been fastening armaflex round pipe insulation if the gap is large. I have this same issue but my door is metal so I think a thicker seal is the solution for me.