#1 Examine how the roofing felt turns up under the lead, because as posted I believe you may have a problem there.
#2 Repoint the chimney, SBR in the mortar where necessary.
Absolutely do not dry-line or the moisture behind will produce a string of problems with floorboards, joists, track through any party wall nearby - trying to hide the problem is a recipe for a total disaster that should be obvious and worries me that they should suggest it. Makes me want to examine that roof more closely.
BTW, you can get wireless and wired cameras quite cheaply - it could be useful to stick one in an appropriate position. From the suggest of
*wet plaster* in moderate rain I really do wonder about that flat roof re funneling water.