I have been using tubeless radial tire's since the late '60's on vehicles ranging from 1957 T-Birds, '58 and later pickups and large trucks up to and including eighteen wheelers. The flat tire's and/or problems I have had can be counted on just a few fingers.
Generally speaking I had more trouble with tubeless tires with tubes in them, than those without tubes. On a few very rare occasions, a tube was put put into a tubeless tire, the tube generally failed causing the tire to fail.
My '36 Ford is the only vehicle I own that has bias ply tires on it.. I don't drive it much, the tires on it are over 18 years old. I'm planning on finishing up some maintenance work on the '36 this year. One of the items will be the tires, which I intend to replace with tubeless radials on the stock wheels.