Consider first a hypothetical situation in which no species, sub-species or variety had ever gone extinct, so every twig on every branch of the tree of life had living representatives. Apart from the mammoth task involved in catching and classifying the resultant billions of 'species', how could we possibly classify individuals and place them into a specific taxon? It would of course be impossible because in reality we would have a mass of living things with no sharp delineation between groups of individuals having characteristics in common and distinct from all others. The dividing lines would be so blurred that we could not easily identify any boundaries. It is only because of regular extinctions and gradual evolution leading an archaic form to change into a modern form across the entire population range, that we can see living things as forming distinct taxa in the first place.
We can see the present but the past is invisible to us directly and can only be inferred from evidence like genetics, morphology, physiology, embryology, with the occasional fossil lending support. It's as though the tree of life were growing underground, with only the very tips of the surviving twigs showing above the surface. And of course, that's exactly where the few fossil remnant of the tree are - in the ground.
The problem with this tree of life model is its simplicity. It is, after all, only a model. It tends to make us lose sight of the fact that a 'species', represented by a single twig on this tree, is not a single entity but tens or hundreds of thousands, even many millions of individuals, all of which are actually little twiglets in their own right. At any one point in time, past, present or future, any group of individuals has the potential to be the start of a new twig on the tree. All that is needed is for them to become genetically isolated in some way and then to evolve in a direction which makes them eventually incapable of breeding successfully with descendants of the parent species.
One problem with this over-simplified model is it is easy to fool simple people with it. For example, when creationist pseudo-scientists reassure their credulous customers that no one has ever seen a new species arise from an existing one, it looks like a rational argument against the Theory of Evolution because they think of a 'species' as a single entity which the Theory of Evolution says suddenly 'evolves' into a new species in a single event. The 'no one has ever seen a monkey give birth to a human' syndrome. Of course, the Theory of Evolution only bears a superficial resemblance to this creationist parody and says nothing of the sort.
In a very few instances, a new species can arise because of a single mutation, such as the example I gave in The Good Shepherd's Purse Is Bad News For Creationists. Occasionally in plants, it can arise by hybridization between related species, as with bread wheat. But, for the vast majority of speciations, the process is a slow, gradual one in which a group of individuals become reproductively (i.e. genetically) isolated for a long period of time.
But let's think about those situations where speciation is, or could have been, a single event such as a mutation or a hybridization. Even if you witnessed it would you have knowingly seen a new species arise? Would you regard a single individual as a new species, or just a mutant? It is only after a population has been established, several, maybe many, generations later that anyone is going to know that earth has a new species. By then, the precise location and the precise founder individual will be unknown and unknowable.
I'll illustrate this with a hypothetical example. Let's take a species of monkey living in a large rain forest spread over several thousand square miles. These monkeys will be able to move freely across their entire range so that genes can flow and spread throughout the entire gene-pool and any evolution due to selective environmental pressures will occur in the entire population.
But, gradually, due to climate change or continental drift, or maybe a change in ocean currents, the forest begins to get drier and turn into grasslands, with trees surviving only close to rivers. In other words, the monkey population is broken up into isolated groups which can no longer interbreed because they simply don't come into contact any more. Each group will be free to evolve according to the local conditions in its woodland. Eventually, maybe after a few hundred thousand years, maybe a million or two, these groups may evolve to the point where they not only look different to each other but may not be able to interbreed even if they do meet up.
So where and what was the 'speciation event'? At what point in the process could an observer say, "Hey! I've just seen speciation occur! It happened when...". In fact, we only know that speciation has occurred retrospectively because, according to our rules of taxonomy, failure to interbreed means they are now different species. Maybe if we had been able to examine them a hundred thousand years ago we might have found that they could still interbreed. Maybe we would have found an incompletely speciated 'ring species'.
There was no sudden emergence of a new species; no sudden branching of the 'tree of life'; no mutation which brought a new species into being and no 'macro-evolution' event. There was no event which creation pseudo-scientists proclaim to be impossible and which they claim has never been seen. All there was was a slow accumulation of difference, directed by natural selection with each group doing nothing but struggling to survive and reproduce with the ones which left the most descendant contributing the most genes to the gene-pool.
Now, take the same scenario, only this time the climate changed again after a few tens of thousands of years and the isolated scattered groups could once again mix freely. But this time maybe they had not diverged sufficiently to prevent interbreeding, or maybe one group now had a significant advantage over the others. In these cases, the group with the genes which gave them greater success would come to dominate and possibly replace the others.
Is this speciation? Is this the point at which we can say a new species arose and the 'archaic' form went extinct? Or is this merely evolution of the entire species? Were those groups isolated for a few thousand years new twigs on the monkey branch of the tree of life, or were they merely groups of individuals with the potential to become new species, but which never quite made it? Certainly, the day they came back into contact, nothing happened to their genes. It was not a change on their part which caused them to re-establish contact. It was the environment which changed. And of course we can never know whether they could interbreed or not, so we will never know if they were ever a different species according to our rules of taxonomy. More importantly, what was there an observer could have seen as a speciation event? Again, there was nothing because it was not an event, it was the result of a process.
And it actually makes not one jot of difference.
To a palaeontologist, examining fossils from across this period, if he/she were fortunate enough to find any, it might look like there were different species at one point, and then all but one of them suddenly went extinct, leaving just the present species. Compressing geological time into the geological column where a few thousand years can be a few inches, it might look like climate change caused an extinction with only one 'species' of this group of monkeys surviving it.
A creationist 'scientist' would of course chortle at the lack of transitional fossils and probably write another book about how this has destroyed Darwinian evolution and proved it was the Christian god who made the monkey 'kind' 6000 years ago.
(I'll resist the temptation to point out that someone is making a monkey out of someone - oh! I just did)
To nature, all that has happened is that living things have evolved according to the pressures in their environments. It is the environments which have changed and life has adapted to it. Nature has not the slightest interest in making new species. They arise only incidentally in the process of evolution and usually nobody and nothing pays it the slightest attention because nothing remarkable, or even recognisable happens.
And we only know about it when it's done and dusted.
Speciation is rarely a single event. It is a process spread over a prolonged period and we do not know it has happened until well after the event. Even if, as in the rare case of a single mutation, or a chance hybridisation, we would not regard it as speciation unless, many years later, we find it has produced successful descendants. 'Failure' to observe a specific event tells us nothing about the validity of the Theory of Evolution because the Theory of Evolution tells us not to expect one.