When the cars were designed and built, a smooth ride was the goal. I suspect that if you read the "full load" or "for speeds over 70mph" tire pressure recommendations, those pressure specs will be more like 30psi or even 32psi (which was the max pressure for normal 4-ply rated tires back then).
You also have to understand that road isolation was not nearly as good with the older bodies as it was in the middle 1960s bodies, for example. Or even the '63 bodies. Unibody cars never were the quietest cars on the planet, either, without large amounts of sound deadening in them. Think "undercoat" that might not be there any more, if it ever might have been. So look in the wheel wells and such, plus the underbody for undercoating of any kind.
Steel-belted tires are more prone to transfer road harshness into the body at around 45mph, being quieter at 55mph and higher. Something about the harmonics of the tires compared to bias-ply or belted-bias-ply tires. These things became apparent when radials were first sold in the middle 1960s, especially on some Chrysler products.
It is more the body structure and lack of sound and heat insulation (compared to modern cars) that make things sound noisier. First time I drove my grandmother's '58 Fairlane, it was louder inside than our '66 Newport, back then. Has little to nothing to do with the type of front-end suspension on the cars, but how extensively they were insulated.
In the short term, be more concerned about wind noise from aged weatherstrips. Plus the affect of aged body mounts on things, too.
So, keep the radials you now have. Start with 30psi frt/28psi rr and watch the tire wear. Especially with the narrow 5" rim width of the wheels used back then.
One possible fix to decrease the road noise is a "DynaMat" type underlayment for the floor. It can decrease noise transmission plus add some heat insulation, too. Shop for the best price as most of it is all the same stuff, although some might be thicker than others.
Glad the car is driving again!
CBODY67