It's a difficult call Bazile, and possibly evokes similar feelings as those expressed towards 'modern' and 'contemporary' art.
And were not in the richest of times...and we need to be practical....
My bottom line with buildings (and I love buildings) is, once the purpose for which it was created runs out of steam, emergency lights start flashing.
If a building wasn't fit for purpose in the first place, it's an easy decision to let it go.
Some (only some) post-1960 buildings are both well constructed and attractive, so then it comes down to the local context. Does it sit well, is it well used / well regarded, if conversion or upgrades are needed is it economically feasible?
I don't know the case with the old Birmingham central library, but usually that date of building needs vast amounts of additional insulation, re-wiring and re-plumbing to bring them in line, so demolition is cheaper.
I think the saddest thing about many public buildings done in the last decade or so is that they are intended not to last more than 25 years - it's an assumption that they'll be outmoded and demolished. That strikes me as wasteful.
Also, with libraries and many other buildings, the exterior is irrelevant - it's how they function to serve their audiences that is important. Glitzy architectural schemes strike me as losing sight of that important fact.