Best choice for flat roof material

The most important considerations are preparations and safety.

Make sure to fully evaluate the condition of the rafters at the top of the wall interface point. If the top log(s) are as severely rotted as you say the rafters are likely to be rotted at that point too. If they are rotted then you would not want to be jacking from the outside ends of the rafter tails. It may require jacking from the inside of the wall perimeter. In the end rotted rafters may cause you to have to deconstruct the roof structure so that be necessary rafters can also be replaced one by one and then re-assemble the structure.

You would need to consider adding some temporary cross ties as close to the bottom end of each rafter as possible. This will help to keep the rafters from spreading when you start the lifting process.

There will also need to be bracing added outside the walls from the orange pieces down at an angle to the ground. These braces need to be located both parallel to the rafters and perpendicular to the rafters. These can be anchored to posts dug into the ground. The braces will help prevent the roof structure from shifting laterally as it is raised up off the walls. One way the braces can be constructed is with overlapping lengths of 2x lumber that are clamped together. As jacking proceeds these clamps are loosened and reset to allow for the overall lengthening as the roof goes up and shortening as the roof comes back down.

Before doing any jacking do not forget to ensure that the roof structure is completely free from the chimney. It would not go well if the chimney were to impeded the roof movement.

Lastly ensure that there is plenty of help available during the raising and lowering process.

Definitely do the chimney project first. Any foot traffic on your roof reduces its life by damaging the surface. In the case of asphalt shingles, the granules are the primary weather and UV light protection. Walking on an asphalt roof always degrades the granule surface, especially at the bottom edge of each shingle. That takes years off their useful life.