No relining doesn't weaken the barrel. You have a liner made of modern steels, if memory serves, 4140 from TJ's, often times much stroger than the original barrel.

ReLining isn't for every gun and I suggest it only for maintaining the original barrel profile and markings. Same for a rebore, except the caliber markings need changed.

Most barrel makers can and will duplicate your barrel for a price of course.

I have back in the olden days had a number of guns rebored for customers. Never seen it make a bad barrel shoot better. A good barrel that shot good and got shot out seems to continue to shoot good rebored. In the end the cost of a good rebore seemed to equal the cost of buying a new barrel and installing it. Reboring seemed to be the way to go in the days before we had so many good barrel makers.