More canals than Venice, more trees than Paris, Birmingham city centre is much improved from where it was even 10 years ago let alone 20, south Birmingham is very leafy, if look from the centre south there are very few tall buildings and all you can see is trees, north Birmingham is a bit of a shit hole, but then every City has shit areas.
As for it not being the second city, it has the second largest population after London, over quarter of a million more people than Leeds which is in third place, Manchester only just making the top 10, Birmingham being more than twice its size.
Manchester may have claims to be cooler or trendier, depending on taste, but places like Harpurhey are hardly centres of culture and forward thinking.