Thermodynamic functions, the (higher-order) fluctuations and correlations of conserved charges at $\mu_B=0$, and the Fourier coefficients of net-baryon density at imaginary $\mu_B$, are considered in the framework of a Hagedorn bag-like model with a crossover transition. The qualitative behavior of these observables is found to be compatible with lattice QCD results. Fair quantitative description of the lattice data is obtained when quasiparticle-type quarks and gluons with nonzero masses are introduced into the bag spectrum. The equation of state of the model exhibits a smooth and wide crossover transition.