Should applicants with extremely wealthy families be eligible for merit-based scholarships?

Yes. Just because the parents are wealthy, it does not mean that the kids have access to the money. I know of instances where the parents stipulated that they will not pay for their children's education unless the latter selected a specific major. Some parents who came up the hard way want their offspring to learn the value of money by earning it for themselves.

Merit-based scholarships as the name suggests should be based purely on merit. If the candidate has proved himself worthy of it (to whoever takes that decision) then no other factor should be taken into consideration. You usually see this given as a recognition award or as a distinction to a few really high performing candidates. If you look at the candidate's background and try to assess whether he really needs the scholarship then that is no longer a "Merit-based" scholarship that is a "Need-based" scholarship. Obviously there will some minimum standard in terms of merit that they may require as well, but that is still a "Need-based" scholarship and not a "Merit-based" scholarship.