what you suggest is not enough to make the system really work fairly.
Let's just use the word "good" to describe a halfway intelligent, objective person who has the ability to evaluate the evidence and the guts to make their own decision independently (the other problem with juries is that one or two strong individuals can persuade others to vote against their own instincts).
So what I see happening, have seen as an observer, is that obvious "good jurors" are routinely eliminated. Now you might say that some still manage to get picked, but when you have a 12-person jury where 2 out of 10 might be smart and desirable jurors....well cases get skewed very easily. Even if you can say that 50% of cases are fairly juried, that is not good enough. Lawyers are lovin it and the legal profession fights any attempt to change this flawed system. It would be fairer if juries were simply random. And in lesser civil cases, juries are just a waste of time and money for everybody--Judge Judy is perfectly fine. In criminal cases I see that, in theory the jury system has merit, but not the way that it works currently.
The best way not to be on a jury is to show up and state the inconvenient truth --that the jury system is not working very well. And be shown the door immediately, while the lawyers try to wipe the guilty smirks off their faces.
You're talking about an ideal, not the reality. Without major changes, I could never support the jury system. It is grossly exploited for profit and success by any clever lawyer, as happened in the Trayvon case. But what people need to see is that this is NOT uncommon. Sorry to be promoting this idea re. the Martin case--but people need to know it.
I strongly advise anyone who doesn't agree to go observe some trials in your locality. It will open your eyes.