Why didn't Paul just give up and die?

Paul did not give up because he recognized unfinished business: people still needed to be reached with the gospel as only he could do it. Although death would have been a welcome escape from suffering, his life was not his own; he had long surrendered it, so he labored on.

Les explains that Paul had been on a sickbed in Philippi, had nearly lost his life in Ephesus, and faced constant attack from adversaries inside and outside the church. Death seemed a welcome escape. But Paul realized there was unfinished business: people still needed to be reached with the gospel as only he could do it. His life wasn't his own; he had long pitched that. So he labored, willing rather to be absent from the body and present with the Lord, but not until his work was done.

2 Corinthians 5:8

We are confident, I say, and willing rather to be absent from the body, and to be present with the Lord.

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What did Paul realize that kept him from wanting to die?

Paul realized there was unfinished business: people still needed to be reached with the gospel, as only he could do it. Though confident and willing to be absent from the body and present with the Lord, that remaining work kept him from seeking death as an escape.