The answer to this question can be found in chapter twelve of Tuck Everlasting. During this chapter, Angus Tuck takes Winnie out in a rowboat to the nearby pond. It is his way of getting her alone, because he wants to explain the importance of keeping the spring a secret. Angus is legitimately worried that people would rush to the spring and drink from it without understanding the repercussions of what they are doing.
"If people knowed about the spring down there in Treegap, they'd all come running like pigs to slops. They'd trample each other, trying to get some of that water. That'd be bad enough, but afterwards—can you imagine? All the little ones little forever, all the old ones old forever. Can you picture what that means? Forever? The wheel would keep on going round, the water rolling by to the ocean, but the people would've turned into nothing but rocks by the side of the road. 'Cause they wouldn't know till after, and then it'd be too late."
Angus Tuck no longer sees being immortal as a blessing. He desperately wants to get back "on the wheel." That means that he wants to feel the effects of time like everybody else. He wants the ability to grow old and die. To Angus, being able to die lets him know that he had really lived. Angus believes that if hundreds of people took a drink from the spring, then hundreds of people would also feel like he does.