I can see four questions in your post - correct me if I missed anything?
1. How to get the soap out
2. Does this save time
3. Does this waste water
3. Does this save money
1. If you do a google search on how your laundry machine works, you'll discover (like I did because I had the same question) that the machine does it exactly the same way that you'd do it by hand - fill the tub with soapy water, agitate the daylights out of the clothes to force the soapy water through the clothes and lift the dirt off, and then drain the soapy water away. That's the first portion of the wash cycle. Then the machine fills up with clean water, swishes the soapy clothes around in the rinse water (this is when your clothes softener would be added if you use one) and then drains out, leaving rinsed clothing. That's the rinse cycle. Then the machine spins the snot out of the clothes, using centrifugal force to get as much water as possible before (in theory) they go in the dryer.
From this, you can see that it's identical to what you'd do by hand, except that with the machine it's a set amount of time which generally isn't altered. It doesn't rinse them individually, so in theory, they're all being rinsed at the same time, with the same water, so your final point about not getting all the soap out is valid for both hand and machine washing. Conclusion: washing machines are literally only a labour-saving device.
I do rinse my clothes individually under the tap, because I absolutely hate residual soap, and I'm only doing a few pieces of kids clothing at a time, but it really doesn't take that long and I don't think it uses any more water than a double rinse cycle on the machine does. How much time does it take to rinse out a facecloth when you're in the shower? 20 sec, tops? And there's a lot more soap on the facecloth than there is in your clothes. Bear in mind that there's actually very little soap used in each bucket, and you can decrease the amount of soap in your washed clothes by wringing the soapy clothes back into the wash bucket before putting your clothes into the rinse water, and thereby keeping the dirty soapy water out of your clean rinse water, if you don't want to do pieces individually.
2. It'll save time only if you need a specific piece and don't want to wait for a full laundry cycle to get it. Hand-washing clothing is generally not a time saving choice for most people. I happen to do it only because I currently have the few minutes it takes while dinner is in the oven. That may change in the future.
3. I think it uses less water, but that's because I'm only doing very small loads. If you're on a water meter (we're on a well so it doesn't matter to us, cost-wise) it would definitely be an interesting comparison.
4. It saves money because you're not using the electricity to run the machine. That's the only hard savings - you're still using hot water (maybe) and soap.
Hope this helps a bit....