I never owned anything but steel bikes...1960s upright Schwinns, includng a Sting Ray,
a 1976 Azuki 10-speed, A 1987 Schwinn Cruiser Supreme, a 1960s Schwinn Varsity and a 1997 Nishiki Blazer MTB, but they were all relatively stiff.
But I never experienced or noticed frame flex until I acquired a lovely, light 1984 Nishiki International 12-speed. The seat and chain stays are very thin, and the front fork tapers into a nice, thin curve. If I stand out of the saddle while coasting (or stationary leaning against a wall, I can bounce up and down on the pedals and the frame has a noticeable give and take. And under heavy pedaling uphill the front of the rear wheel will deflect a quarter inch or so towards the left chain stay. I briefly put a rack on the back last year using the gap by the rear drop-pit as a mounting point. But the rack stiffened the frame and the bike lost most of its 'bounce'.
I bought a 2015 Charge Plug skinny, steel tube framed "distance" bike in 2015. It's between a road bike and a touring bike with regards to wheelbase and stoutness. (They switched to Aluminum in 2016). While the frame looks only slightly beefier than the old road bike, it is indeed stiffer, and the uncurved front fork (for disc brake accommodation) has no disccernable give.
Last year I bought a 2007 Dahon Boardwalk 20-inch folding bike. It's steel, which is heavier than the aluminum frames, but desirable for longer frame life, especially in a folder. Ironically, the only flex on that bike is the super long, but kinda thick aluminum seatpost...I think it's aluminum.
All the bikes get light road tires with flexible sidewalls. I use tire liners for better puncture resistance (all tubes). In the wInter the MTB gets stiffer studded snow tires. On the Plug (700x35), the Blazer (26x1.85) and the Boardwalk (20x1.75) the lighter tires at slightly less than max psi really smooth out the ride (plus the Blazer seat has springs).
The Nishiki International gets 700x28s at 85psi, but on the International the frame provides most of the flex. When I first got the bike it had 700x24 tires and I ran them at 100psi and while the ride was harsher, there was still this lovely flex.
My wife has an aluminum "comfort" bike with 26x1.75 tires, a suspension fork, suspension seat post, and a seat with springs. It all works to dampen the bumps, but it just feels numb to me.
I'd rather have the organic flex of a thin-tubed frame.