Well, first of all, the short period comets indicate the existence of the Kuiper belt. The LONG period comets indicate the existence of the Oort cloud.
The principle of deducing either population of comets is the same. You count the ones you see, you apply basic physics and basic statistics to the sample, and it allows you to estimate the population and distribution of the unseen comets. When you do that, you see that they separate into two distinct populations. You can also see how the populations are distributed. What it tells you is that there is a population of nearby comets that lies more or less in the ecliptic plane, and a distant population of comets that is distributed more or less isotropically.
The difficulty with measuring the Oort cloud is primarily that the sample size is small, and that the distribution of velocities is also compatible with the hypothesis that the long period comets are interlopers from an interstellar population of comets. (The data favor the Oort cloud hypothesis, last I heard.)
The existence of the Kuiper belt can only be avoided by postulating a contrived distribution of short period comets (i.e., that they are arranged so that an anomalous fraction of them come close to the sun, and that an anomalous percentage of them have passed through during the period when we have been watching most intently). You are of course entitled to doubt that short period comets come from the Kuiper belt, just as you are entitled to doubt that rain comes from clouds.
What I can't explain is what fundamentalist Christians have against the Oort cloud and Kuiper belt. You aren't the first I've seen on that jeremiad. Do they run afoul of some obscure Biblical verse, or is it just a general anti-science target of opportunity?