I disagree. If you know how to play a champ (what its abilities are, combos, etc), you should be able to play it well, or at least well enough (I'm talking going even on kills, being beneficial to the team). 85% of this game is the same regardless of what champ you are playing. Since champs mechanics don't usually drastically change, playing any single one once every few months and keeping up on the patch notes should be more than sufficient to maintain a reasonable skill level.
1310 games over 6 seasons equates to half a game per day. Many people are able to play multiple games per day, especially on weekends. It's much more reasonable to assume 20 or 30 games with each champion on average.
Keep in mind here, I'm not talking about playing at master-level carry-the-team can-do-no-wrong. I'm setting the bar at a reasonable, average ability to play. It mostly comes down to making good plays and working well with the team. Focusing on a few select champions doesn't necessarily make your playing any better, it just makes you familiar with that set. Playing all the champions doesn't make you worse at any of them - a lot of them you can group together as similar characteristics (adcs are a good example) Teemo, corkie and MF all have about the same range for their autos, morgana and lux have the same style skills in the same places, etc.