We reconsider constraints on the eigenvalue density of the Dirac operator in the chiral-symmetric phase of two-flavor QCD at finite temperature. To avoid possible ultraviolet divergences, we work on a lattice, employing the overlap Dirac operator, which ensures the exact "chiral" symmetry at finite lattice spacings. Studying multipoint correlation functions in various channels and taking their thermodynamical limit (and then taking the chiral limit), we obtain stronger constraints than those found in the previous studies: both the eigenvalue density at the origin and its first and second derivatives vanish in the chiral limit of two-flavor QCD. In addition, we show that the axial U(1) anomaly becomes invisible in susceptibilities of scalar and pseudoscalar mesons, suggesting that the second-order chiral phase transition with the O(4) scaling is not realized in two-flavor QCD. Possible lattice artifacts when the nonchiral lattice Dirac operator is employed are briefly discussed. DOI: 10.1103/PhysRevD.86.114512