"street:old_name" is not a tag that is in use anywhere. Adding an "old_" in the middle of any tag is very unlikely to be supported by any tool.

It is the name of a street that no longer exists (in the place this highway is located).  The proper tag, if you want to keep the "street:name" intact would be "was:street:name". 

As the street itself doesn't exist anymore and the footway is the sole thing remaining, "old_name" seems perfectly valid to use in this case.