When you contact the city sewer department again you should find out what the written policy is and if there are any exceptions. Have them send you a copy; a lot of times the original policy is a little fuzzy in the minds of those who should know it well. Many places the owner is responsible all the way to the main line however in some places if the sewer is very deep there may be excecptions and the city might take responsibility.
There are lots of places in the country where the sewer systems were put in sixty or more years ago and even the mains are starting to fall apart. If you live in one of those places and you can keep your system limping along long enough without major repairs; there is a good chance the city will be replacing the main before you have to lay out any big bucks. In that case they would be doing the repair anyway.
I really like the other suggestions here about how to kill the roots causing the problem. Maybe you can figure out which tree they are coming from and have it removed. I had a water line broken by a tree roots in our front yard. My wife didn’t want me cutting it down. I am worried that a few years from now we will be facing the same problem again.