A 28 really isn't all that much tire, at least for my purposes, which sound very much like those of the OP. I have 37s on my all-rounder, and they come in very handy on looser gravel, dirt and rain-softened turf. And they're fine on pavement, both bad and good. So I would recommend a cross bike or a light tourer, by any reputable maker. Hybrids are fun, too, not as quick or versatile as a cyclo-cross, but usually much cheaper.
Racing-style road bikes are too specialized, too delicate and too expensive for me.