To do it correct, I would agree that you'd remove the wall after excavating behind it. Just removing the wall and working quickly may or may not work depending on what material is behind the wall. If it is a dry sand, once you remove the wall it will likely just run out. Since the area is above the ground level you may not have much saturation in the sand. Think dry vs wet sand castles. A cohesive soil like clay would give you more time to work and would be self standing for a little while. You may need to do a test dig to determine the soil down there.
If you are on the cheap, you can build a wall in front of the existing wall in any material you like and it will still work fine. The new wall must be able to support the soil as if the existing wall was not in place. As mentioned above, a landscaping block wall or cinder block (CMU) wall would work well, but you could even get by with using the 6x6 as originally used as long as you use the proper staking. Stacking them in place without stakes/anchors will not work! Once the new wall is in place, just remove the top timbers that are bad of the existing wall put compacted aggregate down. If the top timber is still in good shape, still remove just that one so that you can add the aggregate....otherwise this timber will show and give the look of a double wall. You can then pave over the rock or just leave the rock as "trim" or a shoulder.
Leaving the existing wall does have potential risks, as further deterioration will cause settling in those areas. You may find yourself adding more rock or asphalt in these areas to build it back up every few years. Structurally, I see no risks in leaving the existing wall in place.
Also, from the pictures it appears like the tallest part of the wall is in good shape and the sides (shorter height) are the deteriorated part. You may be able to just replace those bad timbers on the side, since they don't have much retained height.