In theory, your idea is great, but in practical terms it is a big mistake. The reason; sanitation.
I was brought right back to Alton Brown's infamous homebrew episode, where he encourages dumping the near boiling wort into your fermenting bucket already full of an equal amount of store bought ice. His realization was that combining the two mediums will result in immidiate cooling, and will quickly get you in a pitchable range. His idea is logical. Again, it is a great idea, but completely ignores sanitization.
There is no way to know what is lurking in the ice or cool tap water that you are pouring your wort into. One of the functions of the boil is to kill off "nasties" in your water, which is why in ancient times, it was safer to drink beer than water. Obviously we are past a point that we need to be concerned about the safety of our drinking water/ice, but that does not mean it is sterile enough to add to cooled wort and still have a healthy, clean fermentation.
This was always one of the things that puzzled me about partial wort boils....I never understood how it could be sanitary to add tap/bottled water to your wort on the way to fermentation. I guess it is a 300 level concern for a 100 level process. I do not want that to sound insulting, I just mean that perhaps at that stage in your process, there are still other things that are being worked out, and that is fine.
That said, do what you need for now, but if your system allows full wort boils, do them. If not, make that your next upgrade. Sanitization is the first easy step toward better beer, and I just don't think it is wise to be knowingly doing something that puts sanitization in question. Ideally, preboil your "topping off" or "chilling" water for 15 minutes or so, then put it into the fridge, covered, to cool before brewday. I guess that would be the best compromise I can offer and still make your idea work without effecting sanitization too much.
Joe