If you are using flash, then you are not going to see much improvement in shot to shot time.
The flash uses high voltage and it takes some time to recharge the flash capacitor, no matter what camera you buy.
Three seconds shot to shot time is about as good as it gets when using flash.

The shutter lag time can be improved over you older cameras.
The Canon SD940IS would be a good choice.

Low light performance (without flash) continues to be a problem with most small cameras.
There are a few cameras that are better than the rest.
The Fujifilm F200EXR is one of those few cameras.

One of the newest cameras on the scene seems to have it all.
The Canon S90 gives you low shutter lag (0.2 second) and better low light (without flash) performance.
For those outdoor action shots it can belt out 1 photo per second when in "Continuous" mode.
Like your old A80 camera the S90 has manual controls.
The advanced photographer in the family will appreciate that.
And it has automatic scene mode selection, which make it family friendly.

Here is a review of the Canon S90:

http://www.steves-digicams.com/camera-reviews/canon/powershot-s90/canon-powershot-s90-review.html