We study quantum chaos and spectral correlations in periodically driven (Floquet) fermionic chains with long-range two-particle interactions, in the presence and absence of particle-number conservation [U(1)] symmetry. We analytically show that the spectral form factor precisely follows the prediction of random matrix theory in the regime of long chains, and for timescales that exceed the so-called Thouless time which scales with the size L as O(L^{2}), or O(L^{0}), in the presence, or absence, of U(1) symmetry, respectively. Using a random phase assumption which essentially requires a long-range nature of the interaction, we demonstrate that the Thouless time scaling is equivalent to the behavior of the spectral gap of a classical Markov chain, which is in the continuous-time (Trotter) limit generated, respectively, by a gapless XXX, or gapped XXZ, spin-1/2 chain Hamiltonian.