I'm skipping the history post. It was dull, even to me.
There's always been periodization in training, but it only got a name in the 1960's, when it got formalized in the literature by Bompa (oops! history). It's worth taking a look at the different types of competitive runners to see who can ignore periodization.
There are racers and pacers and the two types have different goals, different mentalities and thus different training. The separation explains why there's so little overlap between Olympic champions and world record holders. A pacer might go into a race thinking, "To set my PR on this course, I need to average 7:05 miles. There's a long hard hill at the end which will slow me, so I'll try to run even 7:00's until the hill and hope I have enough left." A racer, on the other hand, might think, "One of the two guys I always finish near goes out hard and dies, while the other runs very evenly and has a devastating kick. I'll stay with Mr. Even and let him set the pace and, when we pass the other guy, I'll speed up to create a gap that can't be overcome by a finishing kick." Record setters are usually pacers, who do the same favorite workouts until they feel they're in top shape and then they look for a race. In races like those in the Olympics, one has a definite date to race and one knows the competition, so training is set up to run the best race possible that day. If you look at books written by Olympians or their coaches, they're always training manuals, whereas record setters tend to write personal memoirs about the obstacles they had to overcome: two different ways of looking at things.
If you're training to run a PR in a 5K, you do a variety of workouts, but keep going back to the same ones to look for improvement. When that improvement seems to be plateauing, you race every weekend for several weeks. The first races are for learning (or re-learning) how to race, then one waits until the weather and course are good and one feels right and goes for the PR. After a number of races, your time won't improve and will actually start dropping. That's when you stop racing and go back into training mode. The setting up of periods of training and periods of racing is one form of periodization, though it's not usually thought of that way.
If you're training to run a marathon PR, you can't just jump into a race when you feel like it (unless you're world class), but have to sign up months in advance, so you tend to follow a standard schedule of periodization (i.e., base endurance, hills, intervals, taper). The "pacer" version of training for marathons and ultras would have you sign up for a number of races throughout the year; if on any given race day you don't feel well or the weather's bad, you just make it a long hard training run; if everything clicks, you go for a course PR. By having a race every 7-10 weeks, one automatically adds tapering and recovery after the race and one ends up having a training cycle - which is another form of periodization.
So, it turns out that periodization creeps in, even when you try to avoid it.