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I doubt the seat support was curved initially, intentionally. It probably bent over time, because of weakness of the thin ply, or the delamination resulted in its bending, as you describe. You can likely install 1/2" ply (ridgid) and pad it well. Many dining chairs have a 1/2" ridgid support base, this way.
Padding typically includes 2" (sometimes 3", depending on the seat frame design) foam, a layer of cotton, then the fabric. When I upholster dining seats, I also install a layer of fiberfill over the cotton. The fiberfill allows the fabric to lay and flow over the surface, as it's being stretched/snugged, much easier and smoother, than if applied over cotton. Sometimes, I even install a thin layer of muslin type fabric (a liner, per se) over the fiberfill.... makes for even smoother application of the fabric. Forming and cutting (around corners and other contours) of the muslin, allows for your "learning" how best to apply the main fabric, i.e., allows you to make and learn from any cutting mistakes and fitting mistakes on the liner, rather than making those kinds of mistakes on your fabric.
Some dining seats have coil or K-arc springs, rather than solid bases. These seats are similarly padded, as the ridgid seat supports.
As to attaching whatever ply support: Staple, nail or screw the ply on all 4 corners (and/or between the corners, as well), to help prevent any racking of the framing, especially if these older or otherwise damaged chair frames are a little unstable. As to any leg or bracing looseness, do any woodwork repairs before installing any ply, paddings and fabric.... and even refinishing, if applicable.
Another option, rather than installing a ply seat support, is to install jute webbing, as the seat support. If you elect this option, stretch each webbing strip as tight as you can, across the spans. There's a special tool for stretching jute webbing, but otherwise, try to tighten the webbing as tight as you can, if you don't have the tool. Weave the strands of webbing, across the span, if you elect this option.
I'd like to see the pics, despite my surmising your scenario(s).