With brilliant lighting, if you don't use the full zoom and you are a good photographer, they can produce decent results.
And if they can't, why do they sell them? I was thinking of purchasing a Pentax DSLR when my Sony Alpha gets old, because they seemed even more compatible than the Sony with ancient M42 lens, but the poor average quality of the results I get with the I-10 did not really encourage me (and also the idea of all my old Minolta AF gear becoming useless).
As I am a longtime adept of the "make good images with a less than $100 camera" challenge, I submitted some of my images to the challenges here. They were not rejected, though I didn't win

@trainman1948
I have the same black/chrome I-10, but I did not notice this issue (yet). But I never kept it switched on for a long time. I noticed color balance issues as soon as the light is not quite a standard one, I complained about the camera being unable to select a long exposure in auto mode (could it be solved by a firmware upgrade?), but this gives underexposed pictures.
Does it happen in all modes? Did you try all AE settings? I supposed so, as you are an experienced photographer.

Videos are in M-JPEG, which means they are sequences of JPEG images without any motion compression like in MPEG.
The easiest editor you can find seems to be obviously Womble MPEG Video Wizard DVD. It is very easy, fast efficient, rather expensive ($100), but with one month for trial without limitations, and will bring you very good compatible MPEGs that keep the image ratio.
Yesterday, I tried an H.264 conversion of a 4.30 mn 800 Mb AVI M-JPEG (several clips shot in low light conditions already concatenated with another software) with the free Open Source StaxRip. The conversion was very slow, but I got an excellent 500 Mb MKV file (with even some noise reduction), that displays perfectly on my LG player.

I agree with you, Intheframe: "really for snaps rather than pictures". Maybe this is the first true P&S I use. My first digital camera was a Casio QV3000EX. The overall concept was of a P&S, it had boring menus, but a great Canon F/2 lens, and I could use it like a real photo camera. Even for my tiny Samsung NV3: I used it mostly in "creative" mode, hardly ever allowed the flash, once again like a real camera.
On the I-10 I begin to discover the "for dummies" features: smile detection and so on
One of the main reasons of my purchase was (after the compulsive buying I already alleged) the video recording feature: I use it the same way as you, but for my grandchildren. Videos are fine, though a bit too noisy at low light, with rather poor sound. Which software do you use to edit them? I never install the software bundled with my cameras, but maybe there is something for the video on the CD. As for now the videos I get are not directly readable on a recent LG Blu-Ray Divx certified player which accepts really a lot of formats. I have to convert into HD MPEG.

Who would buy a camera with such a tiny sensor (noise, no bokeh, etc), and a new lens mount?
Maybe Leica could make such an aberration, and count on the snobbery to sell few copies for 20 times the price they are worth. But this is not the niche market of Pentax.

I generally use DSLRs or more advanced cameras, but, as I found it for €99, I confess I made a compulsive buying
The alleged reasons were that the tiny Samsung NV3 I always carry in my pocket was getting old and lacked a true anti-shake feature.

Here is a small rather pessimistic review of my first tries:

Pros:

- Eye-catching design.

Cons:

- The camera definitely lacks a mode selector knob, so you must always navigate through menus with those annoying beeps.

- The zoom command, as I had read in some reviews is ill placed, for you have to release your thumb from the camera body when you want to reach it.

- The automatic color balance is often wrong. The noise removing algorithm is far too strong and I didn't find a way to deactivate it. The "medium" sharpening is much too strong and the camera needs to be set to "soft". If you look to pictures in real pixels size, the grain aspect is rather ugly.

- The camera in "P" mode cannot set itself to a speed that is considered likely to produce a motion blur. So if you disabled the flash, you must (through the menus... beep beep beep and so on) select the night shot option.

- The macro feature is really deceiving compared to my old NV3 (which also got bad reviews when it was released).

I first thought I should bring my model back to the reseller, for it had an enormous dead pixel near the center, visible even on medium size previews. As I noticed there was a pixel mapping feature, and I didn't want to bother with commercial negotiations, I kept this one, but the pixel is not completely masked after several pixel mappings. (Sensor pixel mapping should be considered as a pro.

Despite all the foregoing, I hope I'll take good photos (and videos) with this one, as it generally happens with all my cameras. I always hate my new cameras at first...