This discussion immediately brings to mind all of the iOS apps that become completely unusable the day a major iOS update is released, and then take a week to fix.
I get that not everyone has spare time and energy, but it doesn't take that long to grab a beta, put it on a device (or load up the simulator) and find out. Leaving it until the last minute just forces you into a panic, and seems irresponsible when you had months to just test your product.
Even if you don't use any of the new features, there may be deprecation warnings and such that one should be aware of sooner rather than later. Tweaking early to ensure compatibility with current-LTS and future-LTS versions is worth some cycles: if you don't, you'll pay for it eventually when old(er)-LTS is inevitably retired.