There are a few ways to have good angular resolution with a weird PSF. Even with just optics, you could build an instrument with a PSF that has a narrow central peak with a big ring around it. This would be great for separately resolving nearby objects but bad for resolving a single object against a background of many nearby objects.

For aperture synthesis, particularly strange things happen. If you take a quick (no rotational synthesis) exposure with a huge 3-antenna array, for example, you only get 3 choose 2 = 6 degrees of spatial freedom, but you get very fine angular resolution. In practice, you do "map making", in which you try to extract specifically the parameters you care about, but if you try to make an actual picture, you certainly can't fill in your whole field of view with tiny pixels, since you can't get past the small number of degrees of freedom.

My personal favorite example of aperture synthesis is the images of Saggitarius A*. You can get incredible detail, but the "images" are made under certain assumptions and don't represent actual individual pixels with reasonable PSFs.