I realise I'm late to this party, but it might be worth mentioning that I have that bog-standard cheapo VtL adaptor from eBay. I have run a 3 kw kettle from it many many times, and it boils just as quickly as it does when plugged into the mains. I have also run an 800 watt microwave which nominally draws 1.2 kw, and an induction hob rated at 2 kw. I also run smaller stuff like a pump for an airbed and the charger for my e-bike. I once accidentally ran both the kettle and the airbed pump at the same time.
I've mostly done this while out camping, but a couple of times I have run an extension lead into the house and run the kettle and the microwave from this. (One at a time, obviously.)
I haven't had any issues at all and everything seems fine. I'm currently having thoughts about maybe running the extension lead into the house to run a 2 kw electric heater during the day, on 3.5p/unit night-time electricity, given the price of fuel oil this month!
Obviously this has its limitations and is likely to constitute a trip hazard, but it's a hell of a lot better than nothing in a power cut. I think I could probably run the wifi router, my computer, a table lamp and the TV together, continuously (heating is otherwise catered for), and maybe unplug a couple of things to run the kettle or the microwave or the induction hob. I could also plug in the fridge and freezer overnight, I imagine, and if I only opened them momentarily during the day, stuff would probably stay fresh for the duration.