Heh. You just described how jury selection works now, almost exactly. Only the numbers are different. The no-questions-asked strikes are called "peremptory challenges." You can also get a juror stricken if there's a compelling reason: this is called "challenge for cause."
I thought under the current system people were allowed to use their peremptory challenges on individual jurors, rather than on groups.
If one starts with a pool of 18 jurors and gives three peremptory challenges to each counsel, I would suggest the composition of the jury will look less like the community at large than if there are eight half-juries and each counsel strikes three of them. Even though in the latter case, counsel would be striking a larger percentage of the jury pool, the composition of each half-juries would likely be closer to the norm than the stance of any individual juror.
For example, suppose that the jury pool was 15% black. In a pool of 18 jurors, there's a 72% chance that there would be three or fewer blacks in the pool, thus allowing their complete elimination. By contrast, because 62% of the 6-person half-juries would have at least one black, there would be about a 64% chance that at least four of the them would have at least one black; three peremptory challenges would not suffice for eliminating all of them.
Even though in the former case each lawyer strikes 1/6 of the pool is eliminated while in the latter case each lawyer strikes 3/8, the latter approach would allow much less alteration of jury composition.