Is it realistic to hope for an ICC profile that allows you to do no color correction?
If not, is it realistic to hope to find a set of adjustments to make that should generally work for all your images?
Or, does it still come down to having the original next to a calibrated screen and messing with it?
No it is not realistic. Profiling your scanner is important and will give you a very substantial head-start, but it generally does not eliminate the need for pre or post-scan image editing. More often than not you will need a combination of good profiling and good image editing to achieve a good result.
And no, it does not necessarily come down to having the original next to your screen and "messing with it". It depends on what you are trying to achieve. With well-conceptualized, purposeful image editing, I can pull shadow detail out of dense Kodachromes that were barely perceptible in the original media, and I can correct colour casts in a scan etc., etc., to make the final result look much better than the original media. In fact, much of the time, once I've produced a scan, I just focus on the digital image. I only revert back to looking at the original in case of real need to see a reference colour, for example. But there are instances when what you want is a faithful reproduction of what the original looks like, and then you do need it beside you, on a light table.
I go into a lot of detail on these matters in my recently published book on Scanning with SilverFast 8. Quite a bit of what I have in there is generically useful, but the instruction sets for the scanning work are specific to SilverFast.