We've been living here for a year and are still seeking what you describe. We are in coconut grove and I guess it kind of fits the bill for the most part. There are lots of young families here, we've got a great public library with an incredible children's librarian. There is free yoga at Peacock Park once a week, free bike tours around the Grove through the BID office, two farmers markets (one new on Thursdays and the other on Sat.), the requisite hippy African drum circle made up of mostly aging white people, plenty of good organic food at the small stores like Milam's and Fresh Market. We've got the terrific CG sailing club that's super down to earth and not "yachty" at all, there's a great independent book store, good (non chain) restaurants, and a mix of funky stores that are still hanging on despite the high rents. But progressive approaches to education?? Unless that's happening at the private schools in the area, it's not happening at the public schools here. It's been a big issue for us. It's not like N. Beach Elem (which is actually mid beach--kind of confusing, I know) which is predom Jewish. I hear they have a really incredible curriculum. That area was our second choice and we still consider moving out that way. Diff vibe but still appealing. South Beach (esp South of Fifth (Sofi) is super expensive but more families I hear are moving out that way and squeezing into little condos if they can afford it. That elem school is pretty incredible but I hear you do have to contend with the "moms with bags" and a bit of a snooty element. Otherwise, incredibly progressive education at that school from what I've been told. Definitely check it out if you can afford it and don't mind condo living and/or having no or little yard space.
Miami is also not as diverse as I was hoping it to be. As a northern CA transplant, I was a bit shocked by how segregated it is (esp for Black Americans). I know it's the South but I had been told, Miami was "different." Well, it's not that different when it comes to Black American upward mobility. There is no substantial Black middle class in Miami like there is in other major cities. It's a bit upsetting to us as an interracial family. I hate the fact that while I am cleaning my yard, I sometimes get approached by passerbys wanting to know how much I charge. Blacks live for the most part in the "Black Grove" or what some forum folks call "the bad part" of the Grove. We like the Grove for the most part but are not happy with the education. We do like that there is a good sense of community and people WALK and bike more than any other area I've seen yet. It's got an urban vibe but still so tropical.
We're still looking for that neighborhood you're describing b/c it's really too expensive in the Grove (which is what led me to do my google search that landed on your question.) Parts of South Miami near Sunset is also another option and Pinecrest too but they both have a low walk score in relation to the Grove or Miami Beach. Pinecrest has an awesome super kid friendly community center and although it at first seemed like the god awful suburbs we hate so much, it's growing on us because the pub schools are incredible, there are lots of terrific bike trails, wonderful organic food options, and each time we're at the community center, I run into more Black folks taking their kids to classes there than I do in the grove. I was told Pinecrest was not diverse at all but I've met more progressive (more so than just bleeding heart liberal) there than in the Grove. Ha! So go figure. Appearances are not at all what they seem.
Hope this helps some...it's a complicated city and I'm not quite sold on staying but we'll give it another year before we make up our minds. It's about a good decade behind on environmental policies than most places like Brooklyn and Oakland. There is no substantial recycling at either of my kids schools. Very few people I know compost. There is no compost bin/pick up like in Oakland/Berkeley. The public transportation here stinks for a major city. There are far too many cars on the road and incredible road rage (another thing you might want to consider in choosing a neighborhood--Coconut Grove is situated in a great spot and we can take a number of side streets to avoid the mess of traffic that seems to always be on US1 and Bayshore. Miami does after all have the greatest incidents of pedestrian fatalities in the US--a testament that people don't know how to share the road. In other areas, you're just stuck in your car for a long while. It's far from being Green, progressive city but I have finally met like minded folks here and they are trying to make things change for the better here---slowly.