Ignoring the difficulty of running a #6 wire from the double ground rods to the panel....Add a #6 copper wire as a grounding electrode conductor from the two ground rods 8 inches apart around the house to the GEC from the panel to the other ground rods. Now the green #10 equipment grounding conductor becomes legitimate in that it ends at a point on the grounding electrode system.
Suppose there is a strong surge that results in 1000 surge amps to the ground rods at the panel. And suppose that the resistance to earth of those rods is a near miraculous 10 ohms. The potential of the "ground"; at the service panel is 10,000V above 'absolute' earth potential. In general the voltage drop away from a ground rod is in the first 3 feet of the rod [so the second rod spaced 8 inches from the other distant rod is useless]. The distant rods will be far more than 7000V from the "ground" at the service. The neutral at the service (if installed correctly) is connected to the "ground" at the panel. So the power wiring will be over 7000V from the distant rods - not what you want. (If the attic j-box is also connected to the new service panel the j-box ground will be a fraction of 7000V from the power wires.)
But you suggest a #6 wire connecting the distant ground rods to the new ground rods. A surge from lightning is a very short event. As a result it has relatively high frequency current components. As a result the inductance of the wire is more important than the resistance. The IEEE surge guide (link has been provided) has an example with a ground wire from a cable entry protector to the earthing system that is 30 feet long with 10,000V between ends. A larger diameter wire can greatly lower the resistance, but not so much the inductance.
A nearby lightning strike can also result in the 2 sets of ground rods being thousands of volts apart.
Separate earthing systems are a bad idea. You want to combine all the earthing systems into a single system then connect at a common point to that single system. The distant rods should be abandoned and the ground at the attic j-box connected to the new panel (if it is not already connected).
The distant ground rods increase the risk of damage.
I agree. The connection to a ground rod only has to be #6 wire because a rod is a pretty crappy electrode. Ufer grounds only have to be #4. The wire size to connect a metal water service pipe goes up as the size of the electrical service increases.There is a table of sizes for GECs that is dependent on the amperage of the electrical system. The maximum GEC required going to ground rods is #6.