Yeah Navier stokes accounts for continuity over Burgers' which elevates you from what might be conventional game-dev like water to ANSYS grade CFD. Although, realistic CFD has its tradeoffs too. Solvers like LF, HLLE, and HLLC all offer computation vs. realism tradeoffs. LF is branchless, but struggles with certain sonic/supersonic shock wave characteristics (which one would see in compressible flow only anyway). For incompressible flow I'd expect the final visual realism to be in the order of Burgers -> LF -> HLLE -> HLLC [1]. The vast majority of the industry enjoys HLLC for mechanical/civil engineering, but I'm often fascinated by just how much one can cheat to get realistic incompressible/compressible flow. You can even further hamstring Burgers' and be left with something resembling the wave equation [2], which is the absolute cheapest "CFD" available.

[1] https://en.wikipedia.org/wiki/Riemann_solver

[2] https://en.wikipedia.org/wiki/Wave_equation