The 16 was popular for a number of reasons. I have an old American 16 Gauge SxS, and it handles pretty well for what it is. The 20 Gauge with the old vegetable wads were much more limited than today's 20 Gauge ammo is, and The 12 Gauge version of the gun is a P-I-G pig.
The 16 packs a punch without feeling like a fencepost.
The 16 was the do-it-all gauge, useful for every kind of game, and the guns typicallly felt better in the hands than 12s. Many still do. Higher-end SxS guns tend to be available in 16 Gauge to this day. It's really the perfect bore for pheasant hunting -- but that makes it a specialty gun, not the all-purpose gun it once was.
A few things led to the 16's massive decline (never say the 16 is "dead" because it just draws out a bunch of fanatics who claim it's not dead because they have a closet full of 75-year-old guns chambered in it, or because Browning releases a few SHOT Show editions in 16 now and then
). As oneounceload said, it's not a skeet class at this point. The non-toxic shot laws deep-sixed it more or less, since 12 Gauge had a 3" standard and 16 does not, and steel is less dense than lead. The 20 Gauge 3" standard helped, too, since it made the 20 Gauge even more versatile in the field than it already had been. Plastic wads also allowed more versatility in shell loadings.
Is the 16 still a nice gauge? Yes, it is. Remember: shotguns are really about the
gun, more than the bore. Gun handling is top priority in upland hunting, and if you swing an Ithaca 37 in 16, it's obvious why someone would choose it over the 12.
However, when one can shoot 1 oz. well from a 12, as well as from a 20, it does make it hard to justify spending the money on something in between. The 37 in 20 is an awfully nice gun, also. Furthermore, gunmakers have been producing all-steel 12 Gauges, even guns with two barrels, that handle like the old 16s did, and alloy guns that are lighter and quicker than most were. Finally, per rcmodel, some manufacturers have produced 16s on the same frames as 12s, which means they tend to feel pretty close to the same. That again makes it hard to justify adding another gauge to the personal armory.