Rules based on Gestalt psychology could be seen as a 'local minima' in terms of optimal image processing. Some of them could be surprisingly effective, but difficult to extend and improve upon, since they assume certain high level attributes that might not generalize well.
"If it's circular, then it's a fruit 90% of the time"
Modern methods like neural networks come at the problem from another direction, and try to build towards those high level attributes incrementally. You could say that a goal of these modern methods is to recreate a version of those 'assumptions' or 'Gestalt rules' empirically. That way, we can feel more confident that we're not using a locally optimal approach.
"This is the smallest algorithm that will accurately identify fruit 90% of the time"
Are such methods being used or worked on today?
Yes, and not just in image processing. If you look at any industry where fast processing of ambiguous domain-specific data is needed, there are usually systems that were designed by experts in that field using 'hand-crafted' heuristics. A benefit of these approaches is that the 'Gestalt rules' are understandable apart from the system that uses it, and is therefore easier to trust.
Was any progress made on this? Or was this research program dropped?
Aside from trivial cases, employing the gestalt approach in a particular domain requires expertise. "Image Expert" is an expensive and hard to fill role. Even then, you're bounded by that expert's ability to effectively generalize an extremely complex problem. Over the past decades, machine learning has become a lot easier and less expensive to employ, and has been slowly replacing all those 'hand-crafted' approaches.
However, one of the interesting things we're finding when we let the machine learning approaches loose is that they sometimes don't agree with our previous rules. Why they don't agree, and what this means about automation and our psychological biases, is one of the big open questions in AI.
You might also be interested in Saliency Maps, and similar techniques intended to visualize the general rules used by neural networks: https://en.wikipedia.org/wiki/Saliency_map