The only shoes I've found that comes close to what I think 80sTourist wants is something from the BMX group. Minimal tread and 'walkability' that are generally used on a large pinned platform pedal. Vans, Nike, and Adidas are some brands to look at. I don't know if they'll pass the OP's stiffness test.

Brad

PS I last used toe clips in 2003. They were a set of Shimano 600 pedals with a chevron shaped cleat that the shoe's tread fit into and then the straps were tightened.