First of all, not having a vote doesn't necessarily mean that the government elected by the so called rich would suddenly stop protecting the rights as enumerated in the Constitution. That applies to all, regardless of whether they vote or not.
[mildly incoherent rant]
If welfare recipients couldn't vote, the cost of government would be lowered enough that businesses wouldn't need to take drastic measures, like offshoring jobs to be competitive. Thus, the poor wouldn't be poor for long. But that's the problem. Welfare recipients are by and large, uneducated. They don't understand that the offshoring of jobs is not the offshoring of salaries. It's the offshoring of high costs created by heavy tax burdens, created by their actions. Voting for the guy who doles out the most entitlements is a sure way to find yourself desperately in need of those same entitlements. But while receiving said entitlements, you become lazy. So you vote for the guy who is gonna "give you a raise." Then, people like me, who have been steadily employed their entire lives get fed up...because our rights aren't being protected by the government we pay for while the bum up the street gets a free ride on our dime. I suppose, being a died in the wool socialist, you'd classify me as rich. Well, I don't feel like my rights are being protected. I feel like my rights are being trampled on to grant someone else a privilege! So by your argument, the government is already illegitimate!!!
[/mildly incoherent rant]