The leader's problem is that he doesn't understand what a species IS. And that was the whole difficulty in working out the theory of evolution in the first place.
Evolution, as the idea that a species could change, adapt to new circumstances and become substantially different, was the early part of the concept, and it was proposed by Lamarck. But Lamarckian evolution assumed that species originated independently, and multiple species could not share common ancestry. It was only part of the concept as we now have it.
Darwin (and Wallace, who reached the same conclusions independently) noticed a couple of problems with the whole concept of species. One was that biologists weren't doing very well at defining the line between separate species and variations within a species; that often came down to matters of opinion because no reliable answer could be determined. Another was that this concept of separate origins for species didn't explain why species which were unquestionably different so often shared significant characteristics.
Just about every amphibian, reptile, bird, or mammal shows the same basic set of skeletal bones, no matter how differently they're shaped to produce wildly varying bodies and limbs. Similar circumstances show up in the basic characteristics of groups of plants, insects, etc. And the distributions of these creatures also hint at relationships among them. None of this would have any reason to occur, if species could not be related to other species by descent from common ancestors.
A lot of people get stuck, like your discussion leader, on the Lamarkian notion that "different species" are unrelated. Darwin and Wallace ended redefining the CONCEPT of species. In the process, they explained one of the biggest puzzles in the whole classification system, which biologists had been expanding for a couple centuries: why it's a hierarchical "tree" instead of just random sets of strange creatures with no way to organize them into groups.
Gorillas, by the way, are indeed like "cousins" to us; they're NOT ancestors, but humans and gorillas share common ancestors. (We also have a couple of closer "cousin" species, bonobos and chimpanzees. And they're closer to each other than they are to humans.) Our relationship with monkeys is more remote--the common ancestors are further back, before apes in general became separate from Old World monkeys.
That discussion leader, like most who argue against the theory of evolution, depends utterly on maintaining ignorance of the theory in the first place. That's been the creationist approach for a long time, partly because Darwin, in the last edition of "The Origin of Species" (the only one with that exact title), debunked most of the arguments creationists are still making. They cannot allow any of their followers to understand the theory in the first place, or they'd see (a) that the proofs are solid, and (b) that creationist arguments are founded on ignorance of both theory and proofs.
The Galapagos Archipelago, situated under the equator, lies at the distance of between 500 and 600 miles from the shores of South America. Here almost every product of the land and of the water bears the unmistakable stamp of the American continent. There are twenty-six land-birds; of these, twenty-one, or perhaps twenty-three are ranked as distinct species, and would commonly be assumed to have been here created; yet the close affinity of most of these birds to American species is manifest in every character, in their habits, gestures, and tones of voice. So it is with the other animals, and with a large proportion of the plants [...] There is nothing in the conditions of life, in the geological nature of the islands, in their height or climate, or in the proportions in which the several classes are associated together, which closely resembles the conditions of the South American coast: in fact, there is a considerable dissimilarity in all these respects. On the other hand, there is a considerable degree of resemblance in the volcanic nature of the soil, in the climate, height, and size of the islands, between the Galapagos and Cape Verde Archipelagoes: but what an entire and absolute difference in their inhabitants! The inhabitants of the Cape Verde Islands are related to those of Africa, like those of the Galapagos to America. Facts such as these admit of no sort of explanation on the ordinary view of independent creation; whereas on the view here maintained, it is obvious that the Galapagos Islands would be likely to receive colonists from America, whether by occasional means of transport or (though I do not believe in this doctrine) by formerly continuous land, and the Cape Verde Islands from Africa; such colonists would be liable to modification,—the principle of inheritance still betraying their original birthplace.
-- Charles Darwin, "The Origin of Species"; sixth edition (1872)