Ted-
Glad to hear you HADN'T considered using ice in the wort to chill it. I only brought it up because I HAVE heard others going down that path before and wanted to help steer you clear before you (or anyone else for that matter) ran into problems.
RO water by itself has its issues, too, though. If you are doing extract batches, you can get away with it a bit more, but for AG, going pure RO will leave gaps in chemical content that can be detrimental to the beer as well.
In AG, there are enzymes in the grains that activate and chop up the starches in the grist to produce the fermentable sugars.
These enzymes are active only within certain temperature & pH bands. You need proper water chemistry in order for the wort conditions to reach these pH bands. If the bands are hit, the wort won't convert, and you'll have nothing to ferment (other than a starch soup, which the yeast can't use).
Also, hoppy beers such as IPAs and other Pale Ales need sulfates to accentuate the hops and produce a proper hop taste profile. Czech Pilsners have a boatload of hops in them, but the hops taste very smooth and rounded because of the low mineral content in Plzn water.
bitterness and flavor. If you are making low-hopped beers, the difference will be far less noticeable.
Calcium and zinc (amongst other chemicals and minerals) are needed for proper yeast health. You will get a weak fermentation if you don't have enough of these in your wort.
My point was that for extract brewing, you'll generally get a decent beer using your city water if it already tastes decent to drink plain. It will have enough mineral content for the hops and yeast to use without any adjustment. RO water will not unless you add them back in using brewing salts such as gypsum, calcium carbonate, calcium chloride, epsom salt or table salt, or you go with a blend of RO and tap water, which is often an easy and very effective way to do it.
For AG, the pH requirements complicate things much more. AG brewers will often adjust their water or start with RO and build a mineral profile up from scratch to mimic the water content of a particular city (such as Munich, Edinburgh, Cologne, Burton-on-Trent, or Plzn) so that they can produce a more stylistically accurate end product.
I'm not saying everybody NEEDS to go to that extreme by any means, but simply that it is an area that can make a difference for advanced brewers.
Home brewing is one of those things where you can get deeper and deeper into the art and the science of it. That's one of the things i love about it. It can be as easy as boiling up an extract kit, or doing a complicated multi-step decoction mash. The best thing is that you can get grreat beer (or bad beer for that matter) out of either method. It's all up to how far you want to take it.
Have an awesome Labor Day!