As others have suggested, its not a straightforward question. "Obsolescence" can only occur in a given context. Bows and arrows are obsolete as mainline military weapons, but that detracts nothing from their usefulness for recreation or hunting.
One thing that's nowhere near being obsolete in the context of upland bird and small game hunting is the 6-pound gun firing a 1-ounce load. There are plenty of 20s and 12s that fit that description, but for many of us, the 16 is optimized for that combination of weight and payload in a way that neither the 20 nor the 12 can claim.
I'm a fan of the 16-gauge in principle and have a couple nice ones. But it all comes down to the relation between a shooter and an individual gun. Just now, I'm more likely to grab a 20 or light 12 than I am a 16, not because the 16 is "obsolete" but just because my Verney-Carron boxlock 20 is such sweet, handy little wand and I'm getting aquainted with two very different light 12s each a great example of opposite ends of the Darne spectrum--a grade 3 Halifax and a V21 (sorry about fingerprints on bsrrels in pics!).
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So 16 gage guns are still being produced and are available on the used market, factory ammo is readily available, and both guns and ammo function pretty much the way they did a century ago. So no, in the context of recreational shooting and small game hunting, the 16 is not even close to being obsolete, at least not in the sense of 8-track tapes or 5.25" floppy discs.
