Well, there are 54 EU countries and they wouldn't all enter at the same stage, so I think that it's workable.
My general feeling about the Champions League in real life is that, in becoming a de facto super league, it's lost some of its international flavour. 17 of the last 20 winners have come from 4 countries and that's taken the the thrill out of facing top teams from other countries, such as Dynamo Kiev, Red Star Belgrade and so on. Champions League money has become concentrated in these four countries to the point that other countries become virtually excluded. Okay, Celtic had a blip last year but that definitely went against the grain.
With regards to the risk of one club getting into the CL and then dominating domestically, that rather implies that the reverse is true with the current model - and I don't think that it is. Since the inception of the CL, English football has become notable for being dominated by the handful of CL-qualifying clubs, and old greats like Leeds, Nottingham, Newcastle, etc. have fallen by the wayside. I think this is because if there was one team qualifying for the CL, that team has one shot at establishing European glory. If they don't make it - or if their European exertions takes its toll on domestic results - they fail to qualify the following season and another team gets their shot. But with 4 qualifying slots in English football, it's created a merry-go-round where a club can be almost there, can make a slip, but still qualify for the CL, still get that money and hold off a team that might otherwise do better (Arsenal vs Spurs being a good example). I guess I'm tired of every groups stage having Man U, Man City, Chelsea, Arsenal, Real, Barca, Bayern, Dortmund, another German club, Porto and/or Benfica. It's boring.
A one-club-per-nation tournament makes it feel more like a club world cup, or in this case a club European championship. There's a novelty to matches featuring teams you don't get to see very often, as opposed to Arsenal v Juventus yet again. I think that quality might suffer initially, but it will lead to talent being more spread out. Top players will want to play for title-winning teams in any given country, rather than moving to mid-table strugglers in one of the four big leagues. That's as it should be. To give an example, Dejan Lovren should be playing for a team like Barcelona or Juventus - not wasting his time at Southamption, joining because they're able to compete with high wages.
Finally, re: what to do with the Europa League. After going on about how much I like dislike a super league, I think that the Europa League can function as a very effective one. There'd be less teams going into Europe overall - England wouldn't get 1 CL spot and 6 EL spots - but otherwise it can work more or less as it is. I just don't see the point in the CL and the EL both being super leagues - especially when the latter tends to be dominated by teams knocked out of the former. The obvious criticism of this model is that it could result in whoever wins the EL being a better a team than whoever wins the CL - but this could be advantageous, as it would actually give some meaning to the European Super Cup.