This is a very complicated subject and will require a thoughtful decision on your part.
Box gutters are a constant pain. One reason good roofing contractors really do not want to repair them is that they know that eventually they will probably leak again somewhere, if not at the repair. They have a reputation and don't want to tarnish it through a dissatisfied customer. Another reason is that it is not a big money maker for them. They could probably do three or more complete roofs in the time it would take them to repair yours.
The best time of the year to contract this job to a roofer would be during their slow season. In PA it would be in the very late fall (after the really cold weather sets) and before the heavy snows of January (when the ice dams cause problems). You may even want to look into a quality general contractor. Check their references and ask them to explain in detail how they are going to repair the gutters.
My personal choice would be to remove and replace the entire system with new wood all the way around and new lining. I realize cost is a factor, but if done properly you would have many years of service before repairs would be required again. Avoid using the common pressure treated wood, as these have a tendency to walk over time.
Your wanting to retain the look is a capital idea. Here in Pittsburgh, we have many old homes that used box gutters and when people installed aluminum gutters it took away the charm of the house. If you do opt to extend the roofline for the aluminum, do not let them throw a piece of wood over the box gutters. This fix has a tendency to bow over time creating a leaking situation.
One last item. If you go for the repair or replacement, have them line the interior of the whole gutter with a rubber membrane. If properly sealed it should last a good 20 or 30 years. If I missed anything let me know...
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