Just curious to where you find a chasing set. I've always used a tap/die. If the threads were properly formed to begin with, then no new metal will be removed unless the containment is so strongly attached it forces the tap or die to wander. Never experienced it myself, but always learning and growing.
I found mine at Sears, a
Craftsman 48 piece thread chaser set Model # 971 2750. What I like about this set is it covers the most common SAE course and fine threads as well as metric. It also has a thread file for SAE and metric for more severely damaged external threads.
Shop around as the same set is sold under different brands if you want to save a little money.
Summit Racing sells it for less than Sears.
I'm not an engineer but my dad however taught me a few things. Without getting into all the details which I probably would mess up anyway you can look up thread classifications. Basically there are several classifications and they run the gamete from very precise or close fit of male and female threads to less so. Suffice it to say most inexpensive tap and die sets are ground for less precise clearances. That means that when you run the tap into a more precise thread it will remove metal and open up the threads. The same is true when you run a die over a bolt or screw. The parts will still fit but not as nice as they once would. The job of a thread chase tool is to remove debris without removing any metal. They are hardened and will reform a slightly damaged thread but without removing metal like a tap or die.
Like many of you I used to use tap and die sets to clean threads and did notice a sloppy fit afterwards. I have changed my ways though.
UPDATE: I just found that Sears sells two versions of this set. The first one I listed sells for about $80 but they also
list another for under $50 here. The only thing I can think of is that the more expensive set is like the old Craftsman line of tools and the less expensive one is the newer line. I don't know what the exact differences are but they both appear to have the same exact warranty; if it ever breaks take it back for free replacement.