Is one single antibody binding its antigen enough to elicit an immune response?
This is a wonderful question, because it addresses a common misunderstanding.
Short Answer: We have no clear data about the effectiveness of a one single antibody molecule in the immune response of an entire organism, but since many different antibodies can recognize the same pathogen, and . Just as antibodies in a healthy individual are able to find, bind to antigens, and produce an immune response, antibodies from transfusions of IVIG can do the same. They can block, opsonize, activate complement, and regulate B-cells and T-cells.
Long Answer: Surprisingly, brilliantly, our innate immune system is able to generate transmembrane and secreted proteins that bind every possible antigen (or almost every possible antigen, depending on who you ask). See my answer to this earlier question, and the associated general references. We get very excited when we teach this. With our focus on teaching the diversity of antigens and binding regions, it is easy to get the idea that there is a single unique antibody for every possible microbial invader. For students that really take in the problem of antigen diversity and the solution of antibody diversity, this question, or some variation, often follows: how can a single antibody ever do any good? Usually, with some wonder about how to quantitatively match up this diversity with any specific pathogen. This is an example of that question. Here is the answer:
Many different antibodies can recognize a pathogen
A single pathogen has many many antigens, or molecules that, e.g., an antibody could bind to. A single antigen has many epitopes, or specific binding sites for an antibody paratope (the part of the Fab region of an antibody that is specific to the binding site). A single paratope can bind more than one epitope. This is called cross-reactivity. Additionally, a single epitope can be bound by more than one paratope. A good discussion of this principle can be found in Chapter 4 of Frank's Immunology and Evolution of Infectious Disease.
Antibodies are produced in massive quantities
Lauren Somparyic, in How the Immune System Works, estimates that, in the pre-immune state, of the roughly 3 billion B-cells, a single clone is comprised of about 30 cells. Each clone contains many copies of the transmembrane version of its single antibody, and many different clones (as discussed in the paragraph above) will recognize a single pathogen. Once exposed to a pathogen, B-cells that recognize it divide many times over, undergoing a series of changes beyond the scope of this question. At the end of this process, plasma cells, the B-cell progeny that produce secreted antibody, then each secrete hundreds to thousands of antibodies a second. Without ongoing antigen stimulation, many of the plasma cells die, but long lived plasma cells migrate to the bone marrow and continue to secrete enough antibody to allow for the rapid response seen in a subsequent exposure to the same pathogen. The majority of antibodies in the blood of a healthy adult come from these long lived plasma cells in the bone marrow. You can read about this process in chapter 3 and 10 of How the Immune System Works and chapter 7 of Abbas Basic Immunology. Essentially, the circulating antibodies in the $\gamma$-globulin fraction of plasma contain large numbers of each antibody produced in response to exposure to a pathogen. Because of this, we don't have to wonder how one lone antibody in the vast expanse of an organism can be effective in response to many pathogens. We have many, many antibodies that can do this.
IVIG
IVIG is from, as the OP states, pooled samples of antibodies from many thousands of blood and plasma donors. 75% of these are IgG antibodies, meaning they've already undergone class switching and are a result of exposure to antigen. I'm not aware of a good study that clearly catalogues the populations of different clones in IVIG samples, but we can say the set of antibodies in IVIG represent the collective immune memory of the population from which it was pooled, but behave in a similar way to the circulating antibodies in a healthy adult.  You can read more about this here and here. Because of this, the same principles discussed in the above paragraph apply here. We don't have to worry about the lone antibody lost in the massive sea of the human body. Even pooled from thousands of donors, each dose of IVIG contains many, many antibodies that can recognize each important human pathogen.
Transfused antibodies retain their effectiveness
There are a variety of clinical uses for IVIG, but there is clear evidence that the antibodies in these samples retain their effectiveness in their roles as effectors of the immune system. Not only are the antibodies in IVIG effective at blocking and neutralizing pathogen receptors and toxins, but they activate and clear complement, opsonize pathogens and damaged cells for clearance by neutrophils and other phagocytes, control and regulate the B cell population, and interact with the T-cell population, much as they do in the healthy individuals who haven't been given IVIG. The NEJM article I linked above discusses much of this (though it focuses on modulation), this review, which you can access without a paywall, also discusses this.