Well, as someone who's five (I think) years into rehabilitating my feet from orthotics, knowing that it will take more years of work ... don't. Support doesn't come from shoes, unless there's something anatomical going on with your feet like a surgical fusion or a club foot.
I think some cushioning on roads is important, so I'll probably never run minimal even when/if I've regained all the strength and flexibility that normal feet are supposed to have. But basically, rigid shoes/orthotics are like putting your feet in a cast. All the hundreds of bones, muscles and ligaments can't do their work. It just makes them weaker and more rigid, plus it throws off the mechanics of the rest of your body. If your feet are weak and inflexible, stretch and strengthen them.
Are the shoes too narrow for you? If your metatarsals and toes are all squished together, they can't support you. Way too many store personnel want to put me in shoes that are too narrow for my feet, because that's usually all they stock and they want to make a sale.
JMO.....