Howdy!
I know quite a lot about the scientific stuff of cycling, so I can assure you with some authority that Prime does NOT hurt your cycle.
Here's the science stuff:
When we talk about "ammonia readings" around here, we actually are referring to something called Total Ammonia Nitrogen (TAN). TAN is the sum of the Free Ammonia (NH3) and Ammonium (NH4) that is in your water. So NH3 + NH4 = TAN.
Now, the API test kit doesn't tell us how much of each there is. It just gives us the TAN and we know that it is made up of "some amount" of NH3 and NH4.
So if you get a reading of 1.0 ppm Ammonia, that is 1.0 ppm TAN and some of that is Free Ammonia NH3 and the rest of it is Ammonium NH4. We don't know how much there is of what, but it's not important. (there are actually ways to know what the amounts are, but it is not important to this topic, so we will ignore that for now)
With me so far?
OK. So, NH3 (free ammonia) is dangerous to fishies. But NH4 (ammonium) is not. One of Prime's secondary tasks (its first being to remove chlorine/chloramines) is to detoxify the Free Ammonia NH3 in your water by forcibly making it into Ammonium NH4. So since NH4 isn't harmful to fish, it is "detoxifying" the NH3 into a safer form.
The most important thing to know is that the beneficial bacteria will eat EITHER NH3 or NH4. BOTH forms are "food" and will continue your cycle. Scientifically the bacteria does not care or distinguish (much) between them. It will eat the NH3 first, but in the absence of NH3 it will eat NH4.
Also, if a more "official" answer can put your mind at ease, I found this statement from a Seachem employee in their forum archives:
"NH4 should not cause much of a concern, however, it is best to have all ammonia at zero if possible and to use a conditioner such as Prime to detoxify all ammonia,
while letting the beneficial bacteria consume it." [emphasis mine]
So you can see that Seachem here is recommending Prime to detoxify the ammonia while still letting the bacteria eat it.
I hope my wall of text was helpful in explaining how it all works. Now where are those articles so we can re-educate them?