Is a diagonal brace screwed to the post enough support for deck stairs?

That diagonal piece is a knee brace, or at least it started life as one. The idea is legit. Diagonal bracing off the stringer to the post is exactly how you're supposed to stop a stair assembly from racking side to side and taking up that bounce you're feeling in the top step. The problem is the execution. A brace held on with four or five screws at each end is basially decorative. Screws are brittle in shear, and a brace working under load is all shear and tension at those connections. Over a few seasons they work loose or snap off one by one, and the brace just sits there looking like it's doing a job. I've pulled a few of these off old decks and half the screws come out with no threads left on them. The fix isn't complicated. Same piece of timber can stay if it's sound, but it needs proper thru bolts, two at each end minimum, with washers, snugged up and checked once a year. Bolted knee braces like that are completley normal construction and they hold up fine. If the angles are off or the timber has checked badly, cut a new brace and bolt that in instead.

On the grey timber, don't stress. UV turns untreated wood grey on the surface while the stuff underneath stays fine. The test is a screwdriver and a bit of pressure. If it bites in soft or the fibers crumble, you've got rot. If it's hard to punch through, you're good. Pay extra attention anywhere water can sit, so where the brace meets the post and the bottom of the stringers. Also worth checking before you blame the brace. Top step bounce is often just the stringer hanger nails backing out or a cracked hanger, since you're right at the top connection there. Grab the stringer and shove it sideways and downward, see if anything moves at the hanger. If the hangers are tight and the post under the landing is solid, then it's the brace doing nothing, and bolting it properly will likely kill the bounce on its own.

Yeah that reply nails it. I had almost the identical setup on my back deck, a diagonal off the stringer held in with deck screws that did absolutely nothing. Swapped them for carriage bolts, two per end, and the wiggle in the stairs disappeared pretty much straight away. One tip from doing it, drill the holes slightly snug so the timber isn't slopping around on the bolt before you tighten it up. Also went back with a wrench the following spring and gave them a quarter turn, they'd relaxed a bit with the seasons.

Your grandad's generation wouldn't have trusted a screwed-on brace either. Back when I was coming up, we'd let the brace into the post with a shallow notch and pin it with a couple of big nails, or just spike it and call it done. Worked fine for forty years on my old place.

One thing I'd add, once you've got that brace off, have a good look at the base of that corner post where it meets the ground. Water sits there and that's where I've seen more posts go rotten than anywhere else. Grey top with solid wood underneath is normal, but spongy at the footing is a diffrent story.

Same deal here when I redid our stairs a few summers back. Half the old screws just spun in place when I tried backing them out, so the brace was basically doing nothing for years. Getting the drill lined up straight through the post was the fiddly part, but a cheap right angle attachment got me into the tight corner without butchering the hole. Made the whole job a one afternoon thing instead of a weekend curse.