To my mind these insane American rules about washing on balconies are yet another example of how we as a country just don't do apartment living very well. It's all about what looks nice rather than practical use of space. New builds with no storage, open plan everything, no rubbish chutes and not being able to utilise the little outdoor space you have in a sensible way are all part of the same problem
Where has the OP or anyone else said that it's an American rule? 
I was born in 1968, lived in council flats with balcony & rubbish shute until 1976. We then had a council house, with garden and washing line until 1997 (my parents bought the house at some point in time prior to that, not sure when though). I'd already moved out 3-4 years earlier, I bought a leasehold flat, no balcony or outside drying space so I'd use a collapsible clothes horse inside, with windows open. I had mould in every room. After my son was born I would put the clothes horse outside my front door (we were top floor so only the other neighbour used the landing. She never raised an issue, just when she was trying to sell her flat she asked if I'd mind not having the washing out if she had a viewing. No problem for me, I obliged.
We then moved to another leasehold flat (ground floor) with communal garden and 2 whirly gig washing lines for 6 flats. 3 of the flats had balconies, I didn't. Us none balcony owners and 2 of the 3 balcony owners used the outside washing lines. The one flat that didn't used their balcony to dry washing; clothes horse and over balcony racks. I was jealous, wishing I could dry my washing right outside my door.
Each time my mum visited she would make a disparaging remark if these neighbours had their washing out! What does it matter to anyone else? I've asked her why she had a problem with it, "it doesn't look nice", so what do your neighbours think about your washing, on the line, in your garden? "well, we're owners" So are they, ffs!
I know live outside of the UK and it's definitely the done thing to dry outside. Last year we bought a house on a gated development of 20 houses. Only about 25% dry washing outside (we do!), the otgers, I presume, use tumble dryers. It's madness! We have high enough temperatures that mean your washing is dry about 2 hours after you put it out. Why would you pay the electricity and damage our wonderful planet by drying any other way?
In winter I use a clothes horse inside with the doors open, only if it's raining.
Sorry, I know this is a long, rambling post but I just don't understand what is so wrong with someone putting washing outside? I love to see clean fresh clothes/bedding/whatever getting a good drying/airing.
As for my mum, she'd probably write a note like tge OPs neighbour did. Totally forgetting her roots!
OP Hang your washing out, loud and proud! 