I guess it's all up to the definition of the "competitive" word. For me it means "able to compete against other player on the field", for you it seems "able to compete against ANY other deck". My take on this is - as long as your opponents are similarly disadvantaged (trouble acquiring all the cards they'd want for their "dream deck-list"), it's all going to be fine.
Just think of a real world scenario and ask yourself a question - how long would it take you to collect all the cards for the "dream deck-list" in real life, as the LotR was back then (not now, where the old boxes go for $20-$40), so $3.50 per booster pack. Some people were able to buy on average one booster-box per month, and that was a lot. Why should the time be any different online? Once again, for those who enjoy playing their "dreck deck-list" - they have the casual formats, where you basically play "with proxies" - having access to any cards.
I'm all up for rewarding activity and playing, but I challenge you to figure out a way that cannot be easily exploited. The system you have on GCCG is easy to exploit, just find a friend fire off a few games, concede in everyone of them after playing starting fellowship (or even before), and you both cash in the rewards. That's why I don't want to give out rewards just for playing in one-on-one games. Leagues at least require you to play against at least 10 different people each week. And yes, they do reward playing as well as winning, just take a look at the "Overall results" for the league. If people played all the games, even losing all of them, they'd have 40 points (1 point per game), this (currently) puts them on 22nd place, which is 25% of the overall players (84-ish).
It's incredibly difficult to create a fair free-to-play system, as you have to take into account many variables. One very important thing for me, is to make it very difficult (can't really make it impossible) to exploit. And if someone does exploit it, it should be easy to detect that player. This all comes down to - how to structure cash flow into the game and rewards, to reward good play and being active.
It's only fair to give everyone the same amount of cash while providing ways to spend it, which are available only through cash. All the ways you should be able to spend the cash, should be considered an investment - hence the tournament/league entry requiring cash (giving you chance to win product prizes), as well as merchant selling cards (getting you closer to being able to play the "dream deck-list" in tournaments).
Please, I know it's very easy to rant, and point out - what's wrong with a system, please instead try to provide constructive criticism - suggest how to solve the problems.
As for the "catchup time" for new players - with sealed leagues, there is no "catchup time". You have equal chance against other players because you play the same (in size) card pool.
I don't think there is one document that describes the MtGO economy, as it's more of a common knowledge, rather than rules laid down, but here is the official page:
http://www.wizards.com/Magic/Digital/MagicOnline.aspx?x=mtg/digital/magiconline/howdoi