Considering that highway=path is already quite ambiguous, adding just surface=asphalt would be misleading. In a urban environment it should be safe to assume surface=asphalt has excellent smoothness. I personally wouldn't use surface tag at all since half of the year this path is covered with moss/mud/leaves/snow/ice. If you really need to use surface tag then "paved" should be acceptable.

Sure, not actually suggestion to use asfaldipuru or any other nonstandard tag.