How to repair street end of driveway that has loose asphalt?
Asphalt is only as good as the base it's laid on.
The fact that you appear to have vegetation growing out of the cracks implies that the base may be a bit iffy right there.
There's no particular benefit to ripping up 6 feet of driveway and changing to cobblestone if you have a few inches deteriorating - since your picture only shows a small area, and the usual pattern is cracking near the edges, I'm guessing a much smaller area is actually cracked.
Changing to cobblestone does not magically solve anything - cobblestone is also only as good as its base, and is normally a lot more expensive, and it's hard to clear snow from effectively, which is needful in New York State. They can be very slippery and are inherently uneven.
An effective fix in any case will be to remove the damaged area, fix the base so it's solid, and then put (whatever) on top. Whatever could be new asphalt, concrete, or cobblestone. A solid base is excavating down below any traces of topsoil and filling with compacted &quot;road base&quot; - crushed stone including the fine dust, ideally, but that varies somewhat with locale and what's available. Compaction is important. When the base moves, the surface cracks.
One more possible factor is that the asphalt at the edge needs to be laid in &quot;full thickness&quot; - if the base is not cut down at the point where it butts into the road/street surface and you have a thin or tapered layer at the edge, failure is assured.