Exactly his. There has been a large feature creep during these last few years.
Here is our current usage for a summer day without vacuum or dishwasher. I'm having difficulty believing this would require a 9-10 kW inverter, if I'm reading this right.
Apparently it will if you simultaneously make coffee, make tea, heat dinner, chill beer, watch television, do the dishes, and vacuum the floor. With the lights on, because it's night.
One clue is that the total wattage is more than the watt-hours, > 1C discharge rate (if your battery is sized for 24 hours consumption.)
And you need even more watts from the inverter for the surge to start any motor (5x nameplate wattage).
You should be fine with a 4kW or 5kW inverter than can surge to 2x its rating for a few seconds.
1kW of solar panels might peak around 850W in the summer, produce about 5kWh. Lead-acid batteries may range from 65% to 90% efficient (I'm not sure of the exact figure), but if 80% you only get 4kWh to run loads (from the battery). That's short of your 6kWh anticipated consumption, even on the best sunny days.
Put in a lot more PV panels. Newer panels may be 50% more efficient that what you've got, in the same area will produce 50% more power, so consider removing the old ones. If you can put in 2500W of PV, that could run your loads and keep the batteries full.
My PV is about 70% of my battery capacity (but charge rate is regulated to 0.2C), so what's available to operate loads during the day is about 4x the kWh I store for use at night. PV is cheaper than batteries (except for DIY LiFePO4 which comes close), which is one of the reasons I proportioned it this way.
As for that dishwasher - mine has as setting for heating water, and one for no-heat dry. So heating element power consumption is an option for the user.