Is it normal for attic insulation to have a brown dusty coating on it?

What you're describing is almost certainly dust, and the fact that it's bone dry with no smell pretty much rules out mold on its own. Mold needs moisture to grow, and insulation with mold on it doesn't wipe off clean, it usually smells musty and the paper facing or surrounding wood would show staining too. Dry brown film that comes off on your glove is filtered house dust, plain and simple.

I saw the exact same thing in my old 90s house. Warm air from the living space leaks up through every gap around ceiling fixtures, plumbing vents, wiring holes, and the attic hatch, and the fiberglass acts like a filter. The dust gets caught in the fibers and leaves that brownish tint. The giveaway is that it's usually darkest right around those penetration points. Go back up sometime and look at the insulation directly above your recessed lights and along the top plates of interior walls. If those spots are noticeably darker than the rest, that's your air leaks showing themselves. Is it dangerous? Not really, it's just the same dust that's already in your house. But it does mean conditioned air is escaping into the attic, which costs you money every month. If it were my house I'd seal the obvious leaks. Caulk around wire and pipe penetrations, foam the bigger gaps, put weatherstripping on the hatch so it actually seals when closed. Recessed lights from the 90s are often the worst offenders since a lot of them vent right into the attic. Worth checking wether yours are IC rated before piling insulation against them if you ever add more.

One practical thing since you're poking around up there: only step on the joists or lay a board across them. The drywall won't hold your weight between them, and I know at least one person who found that out the hard way. The loose fill on top is fine to leave as is. No need to tear anything out over dust.

Same situation at our place. The brown film was thickest right around the bathroom exhaust fan where the warm air gets pushed past the insulation on its way out, and way lighter everywhere else. I got up there, sealed the gaps I could find with caulk and a can of foam, and when I checked a year later the difference was pretty obvious. The stuff near the hatch still looked clean instead of browning over like it used to. One thing to check while you're up there, make sure your bath fans actually dump outside through the roof or a wall cap. I've seen one vented straight into the attic at a relative's house and that's a whole diffrent problem.

While you're up there, take a look at the roof sheathing too. If you see brown or gray patches running along the underside, especally in lines, that's the same dust story. On my own attic the shadowing lines up with every rafter, 24 inches apart, and I had it looked at during an energy audit just to be sure. Air moves thru tiny gaps in the sheathing and the wood filters it the same way your fiberglass does. Scared me the first time I saw it, cleaned off nothing and smelled like nothing.

Commenting late but this helps me too. My attic is the same era and I noticed the brown film mostly fading to nothing near the gable vent on one end. Figured it was outside dust blowing in through the vent screen rather than air coming up from the house, since the pattern ran backwards from what you'd expect. Still haven't sealed anything yet because I honestly can't tell which direction my leaks go. Is there an easy way to tell whether the dust is coming from below or from outside, or is it not worth overthinking it if the attic stays dry?

Update: went back up last weekend and the brown film was heaviest around the bathroom fan duct, just like one of you said. Rest of the attic was way lighter, and the sheathing only had faint shadowing along the rafters. Sealed the gaps around the light fixtures and the fan housing with foam, so I'm calling it dust and leaving it be.