I do agree with @Dan as to “why” the ubiquitination proteins are arranged the way they are and are able to complete the process together.
It is easier to answer the question after examining the role each protein works.
The main reason is that these proteins vary in structure and selectivity and therefore have different functions in the process on of ubiquitination, thus not one or two proteins are essential but all of them.
Firstly, these proteins are very much different in terms of structure, selectivity, abundance and variation.
To understand this difference lets consider the x-ray structures of these proteins:

Let’s discuss a bit on the functionality of these proteins on named subheadings:
Specificity
The E1 protein (ubiquitin-activating enzyme (E1), a homodimer of ~1050-residue subunits), initiates the ubiquitination process through conjugation by forming a thioester bond. The ubiquitin is then transferred to a specific Cys sulfhydryl group on one of numerous ubiquitin-conjugating enzymes E2.
Most organisms, including yeast and humans, have only one type of E1. Thanks to the role of E1 which play a part in the committed step of the process. Even if proteins were marked for degradation, they would not be processed unless the enzymes system is activated. Basing on structure, I don’t think E2 or E3 would be a good activating enzymes.
The abundance of catalytic cysteine sulfhydryl groups on E2 makes it a good transporting medium for ubiquitin., which make them specialised transporters of ubiquitin.
To expand more on specificity, each of the many E3s present in eukaryotic cells mediates the ubiquitination of a specific set of proteins and thereby marks them for degradation. Different E2 and E3 enzymes exhibit different specificities for target proteins and thus regulate different cellular processes. Some E2 and E3 enzymes are highly localized in certain cellular compartments, reflecting a specialised function.
Selectivity and Cooperation mechanism
In protein degradation, ubiquitin is not just transferred to any protein, in fact proteins only marked for degradation are the ones that get the ubiquitin.
One signal which controls whether a protein becomes ubiquitinated is the half-life of a cytosolic protein mainly determined to a large extent by its amino-terminal residue (this dependency is referred to as the N-terminal rule). Not surprisingly E3 plays a central role in recognising and selecting proteins for degradation. E3 selects proteins by the nature of the N-terminal amino acid.
E2-S ubiquitin transfers ubiquitin to free amino groups on proteins selected by E3, the ubiquitin-protein ligase.
Although the E3 component provides most of the substrate specificity for ubiquitination, the multiple combinations of the E2-E3 complex allow for more finely tuned substrate discrimination.
Faster processing
More than one ubiquitin may be attached to a protein substrate, and tandemly linked chains of ubiquitin also occur via isopeptide bonds.
The attachment of a single molecule of ubiquitin is only a weak signal for degradation. However, the ubiquitination reaction is processive: chains of ubiquitin can be generated by the linkage of the e-amino group of lysine residue 48 of one ubiquitin molecule to the terminal carboxylate of another. Chains of four or more ubiquitin molecules are particularly effective in signalling degradation. These polyubiquitin (polyUb) chains, which can reach lengths of 50 or more ubiquitin molecules, are generated by the E3s (although how they switch from transferring a ubiquitin to the target protein to processively synthesizing a polyubiquitin chain is unknown).
The use of chains of ubiquitin molecules may have at least two advantages. First, the ubiquitin molecules interact with one another to form a binding surface distinct from that created by a single ubiquitin molecule. Second, individual ubiquitin molecules can be cleaved off without loss of the degradation signal.
Summarising the discussion, as we saw since humans have one type of E1 enzymes, if that enzyme alone would have been responsible for the all the processes, then selectivity of proteins would have been very low (recognition and marking for protein degradation or other cellular compartments lacking the enzyme would not have ubiquitination at all). This is one of the reasons why E2 is necessary - thete are several forms of E2 with different specifities (some are about to react with RING domain E3 while others with HECT), which would not be the case if E2 weren't present, E1 alone would not accomplish this either.
E2 is able to recognize the proteins which must “receive ubiquitin”, E1 alone would not achieve this neither the two of them because, they must be first marked by E3.
Now the supposedly unanswered question is since E3 is more selective and abundant wouldn’t it carry the process alone? . Well this is a tough question but I guess the answer lies in the discussion above: Although the E3 component provides most of the substrate specificity for ubiquitination, the multiple combinations of the E2-E3 complex allow for more finely tuned substrate discrimination. These are some points which might address the importance of these enzymes, I believe there are deeper explanations to this but this what I managed to dig in my research.
References