Modern Disney filmmaking (Pixar, WDAS) is much closer to the Miyazaki approach you describe than you'd think. It's very, very iterative; each film goes through 5-6 screenings, during which the story structure can and does change dramatically. People are very definitely working on the final product while the story is still being worked out. One pretty common pattern: with 8-9 months left to go until release, the entire third act has to be scrapped and reworked, and often big chunks of the first and second act reworked to match. Voice actors for the main characters are involved throughout, and often come in many times to record new pages of freshly written dialog.

If all this seems chaotic, it is. It leads to untold stress as the release date looms closer and closer and the ending still isn't figured out, which compresses the schedule for each department to deliver a finished product. Very very rarely, the release date is allowed to slip (see "The Good Dinosaur" for example) but that's really the nuclear option, as it involves shuffling the release schedule and incurs a ton of cost. This is a big part of why these movies cost so much: compressing the schedule means hiring tons of people and paying them tons of overtime.

Source: I worked the better part of a decade at WDAS.