Well, with flash diffusion, yes, surface area is what counts. The thing is that to get that surface area you must also increase volume: if the flash isn't lighting up that whole surface area to begin with, the extra size won't help you.
A good way, if you aren't able to 'light up the room' is to try and bring up the house lights a bit with a longer shutter speed: not enough to make your subject look swimmy, but just enough to let the ambient light come in where the flash is falling off. Just let the flash be a stop or two more powerful than the exposure you'd get without flash: often a little flash will do the job. (Don't take it to extremes, to start, just let that shutter speed get down to maybe 1/60-1/15th of a second and don't ask for a very-stopped-down aperture, as Enoeske refers to: what this does is let the ambient light be closer to and more of the scene's lighting. This will mitigate many vices of direct flash in a lot of situations.)
It's all about balance. There is light in the room, and there is what the flash can do. Think of time and light-per-time: These are essentially adding a very brief flash exposure to an ambient light exposure. If the flash does all the work, you will see pretty much only what the flash does. If you have just a little flash amid a very bright ambient exposure, the flash will be a little brief exposure amid all the rest. (This will look funky if you overdo it, but if you go gently you will still have nice and sharp subjects with a handheldedly-softer background. Bokeh? Wedonneedno stinking bokeh. We brought our own.
A lot of stuff is sold as 'You must light up the room.' I say, no. I say, Learn about 'fill flash,' ....Darn near everything is fill flash. The rest is just quality and degree. And most especially balance. I even keep the pop-up flash turned down on any modern camera.
Oh, but inflatable ones are darn handy, by the way. Not 'the best' but better than the fancy bounce card or giganto plastic Devo hat you left at home, by a long way.