I think it would make sense ... except that modern washing machines are (unfortunately, or a least this has been the case in my experience) almost like designed to break/get unbalanced after a certain number of washes (5-10 years, again, personal experience; my mum's one is going strong for 25 years now - and it was a midrange Philips!)... So the question becomes: do you invest in one that will last your family those 5-10 years, and have the convenience of having things washed when you want them and the ability to take advantage of off-peak cheap energy tariffs, or do you share with several families, and have to re-buy machines every 5-10/(number of people) years?
Laundromats are great... in theory ... except for:
(1) distance to travel with laundry, time waiting for wash (hanging out in the laundromat, even reading, is not comfortable, especially with all of the other people around, including those who want to talk - or chat me up), and time heading home, with wet clothes,
(2) cost - in Oz, the cost of 1 wash at the laundromat is $4 a pop. I do 2 cycles a week (cottons /60*C intensive/ and synthetics/40*C delicate/), which would cost me $8/week. Over a year, that adds up to 52*8=$416. I can get a perfectly decent used machine on gumtree (Ozzie equivalent of craigslist) for $200-$400, and a new one for $450 /my current washing machine is actually one of the few things I bought new and with one of those idiotic extended warranties - I've had one too many used ones die on me, I think because of the wear-and-tear hypothesis listed above/...
(3) washing quality. Laundromat machines tend to be industrial-grade, which (in all but one of the times I've tried them) are quite rough on clothes, and fray jean edges, underwear lacy bits and other sensitive fabric areas - which means in addition to the cost estimate in (2), I'd have to add the cost of buying more clothes due to wear and tear during washing.
Having said that, I have lived in an apartment block which had a cheap shared laundry in the basement with a mix of "normal" and "industrial-grade" high-quality machines, which did not ruin clothes. This was also a block where we paid high body corporate fees, so I think that all of the costs of maintaining the space, machines and their servicing was directly passed onto the tenants. Given my relatively low usage, I am not sure that sharing saved me money...
PS While I agree that dryers are evil, I've lived in enough places in the world to understand that they are essential in certain scenarios (monsoon in India, autumn/winter in Northern Europe in a small apartment ). I completely agree these should be shared.