I think it's important to have realistic expectations of what your camera can and can't do. It certainly is possible to take good, even great pictures at Disney (or anywhere) with a point-and-shoot. In fact, every single picture I took on my last Disney trip was with a point-and-shoot Sony CyberShot (that I ADORE). It was on automatic, I fiddled with no settings in the camera, didn't have a tripod, just worked to find the shots I wanted and I took multiple frames of each picture (when possible).
I don't mean to brag, but I think I got some really, really good pictures considering what I was working with, and in a variety of different situations, like the ones you described:
Waiting to Soar by
Tropical Wilds, on Flickr
Transitions by
Tropical Wilds, on Flickr
Castle View by
Tropical Wilds, on Flickr
Reflections of Japan by
Tropical Wilds, on Flickr
If you want more examples, check my Flickr. Everything that's at Disney I did with a point-and-shoot that (I think... It was a gift) cost less than $200. All the settings were automatic, I didn't use a tripod for any of them, just practiced with the camera beforehand, took multiple frames of everything, and in some cases, just took pictures and hoped to get lucky. Really, with a digital (with the proper memory card), you have the freedom to do that and see what you get later, hoping it all works out.
That said, I was realistic too. I knew on a moving ride, my chances of getting a good picture were slim-to-none unless the ride stopped. In fact, I used my knowledge of the rides to prep myself for potential photo ops on the rides. For example, I knew Snow White had a pause at the end, so I took:
Goodbye Snow White by
Tropical Wilds, on Flickr
And some random on-ride shots on spots with longer straight/head on sections, in the hopes of one or two coming out (they didn't). I also knew on automatic mode, my chances of getting pictures on the truly, truly dark rides (like HM or Peter) or constant motion rides (Splash Mtn) were slim-to-none. I took random shots here and there, but got not really anything usable. Honestly, knowing that actual photographers with the real-deal cameras have a hard time capturing that, I knew I wasn't capturing anything usable and I was OK with that because I just knew that was the limitation of the camera, the settings I was shooting in, and the surroundings.
Now, from a professional photographer standpoint, are the pictures I got with that camera of the same quality? No. But you know what, I'm proud of them and I certainly think that for my purposes (personal sharing and capturing of memories), they are more than exceptional. I look at them, know what it is, and are reminded of a time or a place or a feeling or a trip, and it makes me happy, which is all I wanted. Some of my pictures even went from being kind of blah to just gorgeous after touch-ups in my graphics program (which is the ghetto-est of the ghetto... Not Photoshop or anything like it). Heck, even some pics which I thought were totally unusable due to blur or "noise" became great pictures in their own right after applying processing filters to it (like pencil sketch or watercolor, etc).
So I will say, you can get great shots with a P&S that captures the magic and memories just wonderfully. Mine can, and did, and even though I've upgraded, I still adore my little P&S camera and would never say a bad thing about it. Honestly, seeing what I could do with, both at Disney and at home, and knowing I could do more with some practice, knowledge, and better equipment was the reason I upgraded to the one I have now, which is a more professional-style camera. Even when I go to Disney in September, I'll bring both cameras... I'll sometimes take the larger camera, but I'll always have the P&S on me. Until I get the appropriate lens, my P&S will be my go to camera for macro shots, which I feel it still does much better then my larger camera.
So please, don't get discouraged. I think it's reasonable to say that the fantastic shots you have in your minds eye of pictures in extreme environments (too much dark, lots of motion) will be difficult to get, especially on automatic, but in those non-environmental extremes (night that's still well-lit, ride ques, outdoor during good weather, portraits, landscapes to name a few) you can come away with pictures to be really happy with.
Like I said, if you want more examples, go to my Disney section on my Flickr. All if the pictures on there taken at WDW were done with my CyberShot P&S and there's a lot there I'm really proud of.
And the suggestion on "Understanding Exposure" is a good one, as it's a fantastic book. That said, don't get too invested in it until you get your P&S and know what it can and can't do. When I purchased the book, I hadn't gotten my larger camera yet. I tried to do some of the things he talked about in the book only to find that my P&S simply couldn't do or didn't have the settings to do what he described in the book. The camera was so beginner-friendly that while it had a manual mode, it was built, programmed, and intended to be used in mostly automatic mode, or in the preset modes (dark, light, fireworks, snow, beach, etc) and not all the options/settings he described were available for me to set. That made learning both confusing and frustrating, making me feel more discouraged and intimidated than informed. And honestly, while it's a great book that I like that has helped me figure out my camera, I wonder how useful it'd be to you using a simple P&S in automatic mode.