Flat pedal road bike shoes
For flat pedals, here's my suggested preference order of shoe type:


A skateboarding shoe. They have flat grippy soles with more stiffness than most shoes.
A traditional bicycling shoe (intended for use with clips and straps).
A shoe made for any indoor sport. Indoor soccer shoes (like John Zwinck linked to in a comment), basketball shoes, etc. These will have a flat grippy sole intended for hardwood, that any flat pedal will grip quite nicely. Probably avoid anything "high top".
A shoe made for athletic activities on pavement. Walking shoes, running shoes, etc. These won't be as flat or as grippy as the indoor sport shoes, but will still be quite good.
Any athletic shoe. A shoe made for trail running will tend to have a fairly lumpy sole, but that may still work just fine on your flat pedals.
Any shoe. Face it, flat pedals work with any shoe you can walk in. Preferably something with a secured heel and toe coverage. I don't recommend it, but I've even done it in flip-flops with no problems.


Categories 2 through 5 all have available models that have that sort of futuristic aerodynamic look favored by clipless cycling shoes.

If you're doing long distance riding with flexible shoes and small pedals, you may find you get a bit of a "hot spot" where your foot pushes on the pedal. Stiffer soles and a larger flatter pedal platform can both help with this.

A high-end "BMX" or "mountain biking" flat pedal is much kinder to your foot than the traditional cage pedals that often come free with the bike. To me, the ideal is a thin pedal, just as wide as your shoe at the ball of your foot, with replaceable pins. Those cage pedals have small platforms with a sort of serrated knife edge rim.