AC power doesn't store. It's not like a 0.5kw generator could chug along all day slowly filling a huge... I don't know... tank of water maybe, that you can draw from on demand. The generator needs to provide peak demand for when the coffeemaker and hairdryer and every other darned thing are in use at once.
The rest of the time, the generator is its own "spinning reserve" for in case you turn on one of those loads. It's at full speed but doing next to nothing, powering only your internet router and cell phone charger. That is inefficient, and thus, expensive.
Now your house is a fairly small average load, and may be a candidate for a different way of doing things that allows solar as backup. After all, your 0.5kw is only two 270W solar panels, and with the usual 6x design factor for nights and rainy days, that is still only twelve panels.
The problem is storage, and that requires DC power, preferably low voltage DC, and then you can use batteries for storage. This ideally involves making wide use of low voltage DC for loads which lend themselves such as lighting, some air handling, TV, router, certain PCs, etc.
It's possible to use an inverter to make AC from DC, but the inverter is inefficient all the time, particularly under light load. So you have the same "spinning reserve" problem you have with a generator, where you're paying a king's ransom all the time to have AC power on tap the moment you want it. Better to architect for DC and only spin up an inverter when really needed for a short term use.