The original Constitution itself is silent on the right to vote. Indeed, in deferrence to the godforsaken "State's Rights" faction, the Constitution leaves the determination of voting qualifications to each individual state. So while the 15th Amendment referred for the first time to a "right to vote" and prohibited the states from denying or abridging that right to vote on account of race, states still had significant power to deny that right to vote through literacy tests, poll taxes, and other forms of intimidation. So the 15th Amendment, and all subsequent voting Amendments (i.e. 19th (women) and the 26th (age of 18), are better understood as not establishing a right to vote but instead telling the states that you cannot deny the privilege to vote on account of race, religion, sex, or age. States were free to deny the vote in any other way they saw fit. And many states have tried to make voting as difficult as possible. Long lines, lack of polling stations, shortened registration deadlines, prohibition on felons voting, and harsh residency requirements are all examples.
So a Constitutional Amendment is required to not only explicitly spell out that right to vote, but to take away the ability of the states to deny the previous privilege to vote.