Anything is possible. Question is budget. Fiberglass body on a fabricated slab-style chassis with AWD and excellent suspension geometry would handle like a go-kart with any wheels. But it would be loud, hot, and obnoxious inside.
Large sidewalls will always flex more than short ones - helps with comfort, hurts when you try to turn.
Low ride height helps lower CG. You can change geometry to affect roll centers and make for a very good driving car even if it's not down in the weeds - look at modern SUV's. They no longer drive like dump trucks or threaten to flip over on a gentle curve.
Lots of actual race cars use smaller tires due to class rules, and they turn/handle better than many cars with the 'pro touring' look. Also, in reality, most roads aren't going to support a big *** contact patch and sticky rubber. All its going to do is wear down quick and throw rocks like it's cool.
Sadly, the availability of modern tires in 14 and 15" diameters is probably the biggest hurdle. 17" is probably the smallest one you could get modern tires that will fit the bill. With proper rim and tire size choice, 17"s can avoid the 'big wheel' look, but it's still gonna be bigger than a 14".
14" wheels might also be tough to fit great brakes on, but I bet there's stuff out there which would work better than OEM. Probably not cheap though.
With smaller wheels and tires, the best thing one could do is remove weight. The lighter a car is, the less crucial the rest becomes. The question is how light can you go before it becomes a chore to drive? Less mass often means more NVH (noise, vibration and harshness).
