Thanks TGM, for a thought-provoking question.
I would agree with ag that, on a personal level, I would simply not enjoy my rating to switch without playing actual games. It detracts from the feeling of having 'earned' it.
On a more philosophical level, you can wonder what the objective of a rating system should be. One goal, that you are following, is to make an accurate predictor of outcomes, which by extension is an accurate measure of skill. Ideally, GR would change very little.
I'd argue, on a gaming site like this, that the goal of a rating should not be accuracy, but an incentive for a certain style of play. I would say that the style should be active, competitive and maximising outcomes (meaning solo-driven). Going on this path leads me to other rating systems, that don't necessarily reflect skill at all. As is well known, a downside the of Elo style systems is sitting on high ratings (like some of us, you know who you are), or not taking too much risk playing low Elo players (because, what if you lose?). I for one, would not mind a rating that decreases if you don't play, and increases when you play a diversity of opponents. Neither of which have to do with skill. Also, echoing HR, ratings should be more or less transparent to all users, and with this time factor included, I think it becomes less transparent.
Having said that, I do see the value of trying to measure skill. It may be worth a thought if we should not have two parallel rating systems on the basis of these two objectives.