An on-grid system is the cheapest per unit generated and is the right answer for most urban homes with reasonable supply reliability. It has no batteries and exports surplus to the grid through a net meter.
When off-grid makes sense
Only where there is no reliable grid at all - farm pumps, remote installations, telecom sites. Battery replacement every five to seven years is a real cost that must be built into the payback.
Hybrid systems
A hybrid inverter with a small battery bank makes sense if you face daily outages of two to three hours and want essential loads covered. Size the battery for essentials only, not for the whole house.