Bright light is not an issue if everything is evenly lit. Dark shadows and bright light make for problems.
The options with dark shadows and bright light are:
Sacrifice one or the other (go black with the shadows or go white with the light)
Shoot in RAW and adjust in post
Use HDR or DR
IMO, the auto-DR in the RX100 (and any camera, really) does not produce a good result. My guess is "landscape" and "full auto" modes are using auto-DR.
HDR is better, but auto HDR is not great. It takes some experimenting to find out the right HDR level, and then if anything is moving, HDR has problems (ghosting).
Shoot in RAW, protect the highlights and adjust in post is the only sure-fire way to do it. Remember -- on the RX100 mk 1, the jpgs have at least 2 EVs less DR than the RAW files.
I do not agree that this sensor can't shoot landscape. It certainly can. But the question is always -- what viewing size and what viewing distance. I've gotten VERY useable images from this camera for standard image sizes, but they wouldn't hold up well (possibly, anyway) if looked at 100% or blown up huge. If you aren't blowing up the images, make sure to NOT look at the images at 100%. There simply is no need to do that.
I also find that HSL adjustments are important on separate color channels, even in jpg. Sometimes the pic just needs some selective tweaking that only HSL can provide.
Some samples from my RX100 mk 1:
(admittedly shot in low light at dusk, so no strong sun)
20140315-20140315-DSC05947 by
wt2100, on Flickr
jpgs, using the panorama mode
20140315-20140315-DSC05936 by
wt2100, on Flickr
strong light in the high desert
20140315-20140315-DSC05817 by
wt2100, on Flickr
Not India sun, but strong light with all the snow
20130121-20130121-DSC02865 by
wt2100, on Flickr
Not a landscape, but my favorite pic ever from this camera. Panorama mode. This one I did print out large to hang on my wall
20130729-20130729-DSC04101 by
wt2100, on Flickr