Does a second layer of OSB over a rotted layer cause weight or rot concerns

I wouldn't worry too much about it. The plywood is not typically supporting a load of more than the paper and asphalt shingles that were placed, even more so if you have a considerable pitch on your roof to where it would be difficult or impossible for a person to walk on. It basically serves so that you have something to nail the shingles down securely.

Most newer homes use plywood, I have an older home that used what looks like rough cut deck planks and those are mostly still original. The last time I had my roof replaced, they had to change out a few boards that were terribly dry rotted and some that were badly damaged from previous water damage by the oh-so responsible homeowners who had the home before me. Other older boards look noticeably damaged by the most recent nails but they continue to hold down the shingles and I would still feel safe to stand on my roof.

As long as the shingles are doing their job then the plywood will not be exposed to any water, and as long as your attic has good ventilation during the summer time then you should have nothing to worry about for the next 15-20 years at least.

As a contractor, I will share with you an approach that may help.

First of all, you need to document the current leak with pictures, dates etc. When you communicate with your roofing contractor, be calm and polite, explain precisely the problem and ask to witness his/her inspection. Explain that since this is the second occurrence of the leak since the new roof installation, you are concerned that water may have compromised the insulation and framing and is a potential cause of mold or rot and must be addressed. Tell the contractor that you insist on having an infra-red scan inspection of the effected area to detect any current moisture in the cavity before any more repairs are made. If the roofer is not willing to do this, tell them that you are terribly sorry we could not reach a mutually reasonable solution. Calmly but firmly state that you now have no other choice than to have this scan done yourself and that subsequent repairs will be made by another contractor and the total costs will be litigated in a civil action against him/her. Carefully and concisely summarize your conversation and send a letter with your demands via certified mail immediately after your meeting. If the contractor does not respond favorably after written notification, contact the BBB, and licencing bureau of your town or state. If none of this induces the contractor to act, proceed to having another contractor do the scans and repairs. Follow through with civil action, usually small claims on both the contractor and his/her insurance company.