Life is all about a little compartment controlling the (bio-) chemistry inside it. From the moment the first cellular predecessors started forming in whatever the medium they had was, they had an ‘inside’ and an ‘outside’ and life evolved by taking control of the ‘inside’, protecting it against the ‘outside’ and then by some means, propagating the ‘inside’ to generate a ‘daughter inside’. They key concepts are self-assembly, autocatalysis and self-replication. To briefly introduce them:
Self-assembly describes the spontaneous assembly of a more complex structure from simple starting materials by diffusion. For example, merely adding a zinc salt of a weak acid to a solution of protonated porphyrins will produce the corresponding zincporphyrin. If the porphyrin has an amino function at an appropriate position, these porphyrins will go a step further and dimerise spontaneously in solution. A more simple example that is frequently introduced in high school or undergrad courses are the hydrogen-bonded carboxylic acid dimers that form in aprotic solvents.
Autocatalysis refers to a reaction where the reaction product is able to catalyse its own formation unspecifically. For example, consider the decomposition of cellulose acetate into cellulose and acetic acid. This reaction is catalysed by a proton source (acids), meaning that the more cellulose acetate degrades the more protons in form of acetic acid will be available and the reaction will progress faster. It is important to note that an autocatalytic reaction is typically unspecific: by adding sufficient amounts of a different acid (e.g. HCl), the reaction is also sped up and no rate increase is observable.
Self-replication is probably the most important of the three: it describes the creation of an exact copy of a template from available starting materials. Usually, a self-replicating reaction will also be autocatalytic because with each synthesis a new template is generated allowing for the synthesis of additional individual units. Probably the most well-known example is the replication of DNA where a parent strand serves as the template for an exact copy which becomes the daughter strand. Note that enzymes are perfectly able to synthesise DNA without template—however, unspecifically.
While all three concepts are essential to developing life, the third one is the one that defines life and the possibility of evolution as we know it, because if you cannot generate an exact copy of yourself if you are not passing on yourself to a daughter generation. Furthermore, it becomes essential to restrict any possible replication to the desired self-replication. This is the beginning of the complex relationship between DNA, RNA and proteins/enzymes.
Probably the initial proto-cells depended heavily on the uptake of reactive predecessors to support self-replication from the outside broth but that isn’t a sustainable long-term solution. Thus, as time passed more and more reactions were taken under control to support the proto-cell in what it was doing, more and more simple enzymes developed until finally something resulted that could live on an energy supply and a carbon/nitrogen source alone—the first cells. These have a striking advantage over proto-cells that require the uptake supplies from outside so they probably rapidly prevailed and evolved further.
With every further step in evolution, the machinery became more sophisticated and enzymes became better and more specialised at their job. But more specialised also means more complex and more complex means easier to destroy. Thus, with the intracellular machinery becoming more and more complex it became more and more necessary to regulate every single little detail of what happens within a cell lest there be some way this could be harmful to the cells survival.
For example, a lot of reactions require acidic protons to be supplied to a certain functional group of the substrate. But for a complex cell it would be devastating to generate excess acidic protons and hope they find their targets because in another corner of the same cell an acid-sensitive substrate could be in the making or a basic environment is required to abstract a different proton. In addition, even if there were only one substrate that could react with this acidic proton, that substrate can still potentially react in more than one way with an undesirable pathway being more likely. All this is inhibited if the acidic proton is kept tightly in the centre of a catalytic pocket surrounded by the enzyme’s protein backbone, out of harm’s way.
All of this has lead to quite the opposite scenario from the one you paint: typically, a cell will have an enzyme available for every single reaction it wants to perform and leave nothing to chance or random collisions. Every single reaction that was left outside of an enzyme’s control has the potential to disastrous consequences and the cell is really better off limiting these as much as it can.
To exemplify this, I like to point to one of my favourite examples: Superoxide dismutase. This is an enzyme that catalyses the disproportionation of superoxide $\ce{O2^.-}$ generating diatomic oxygen and hydrogen peroxide. Superoxide is a highly reactive oxygen species that is generated in one-electron reduction of diatomic oxygen. This decomposition reaction is extremely rapid even in the absence of an enzyme: superoxide will disproportionate to the same products so rapidly that it is used as a chemical oxygen generator. However, even that rapid reaction is not fast enough for cells as even the short half-life of superoxide in water might cause irrecoverable damage. Thus, highly complex enzymes evolved and most species have a handful of them to combat superoxides. Furthermore, superoxide dismutases are remarkable in that they have one of the largest catalytic effeciencies (measured as the catalysed reaction’s rate constant divided by the uncatalysed reaction’s rate constant) $k_\text{cat}/k_\text{background} \approx \pu{7e9 M^-1 s^-1}$ – for a reaction whose background rate is already estimated at $k_\text{background} \approx \pu{10^5 M^-1 s^-1}$!
The product of superoxide dismutase, peroxide, is another highly harmful compound that has a high tendency to oxidise anything in proximity so it is also immediately broken down by peroxidases—but at least that doesn’t come as an equally big surprise since peroxide solutions are actually shelf-stable so peroxide has the far more obvious potential of harming half the cell.
Having considered all this dire need for controlling what actually happens in the cell’s inside, it should be less of a surprise that enzymes perform one step only and that each individual step of a biochemical pathway will have an associated enzyme—although it should be mentioned that if a reaction is close to equilibrium then enzymes will often indiscriminately catalyse both the forward and the backward reaction.
In the previous paragraphs I have not at all spoken of the idea of regulation: that a cell can turn on or turn off biochemical pathways as it sees fit. Bluntly, you don’t want to be breaking down your precious storage of glucose-6-phosphate (not actually stored as glucose-6-phosphate) into carbon dioxide and ATP if your cell is already oversaturated with ATP. The multitude of biochemical reactions catalysed by one specific enzyme each allows simple up and down regulation of individual pathways by means of inhibition or enhancing individual enzymes—another immediate advantage for the cell.
Now you might still want to argue why enzymes aren’t collected together to perform multiple steps at once or one after the other. Well—they are. Some of the most interesting molecular assembly lines are polyketide synthases. These multi-enzyme complexes typically consist of individual subunit domains which perform any of these steps: $\ce{C-C}$ bond formation in an aldol reaction typically using malonyl-CoA or methylmalonyl-CoA as a chain extender (ketosynthase), reduction of the resulting ketone (ketoreductase), elimination of water to create a double bond (dehydratase) and reduction of the double bond to give a saturated product (enoylreductase). Among these domains, there is an acyl-carrier protein (ACP) which serves as a robot arm transferring the growing chain from one domain sub-enzyme to the next. After having passed a domain, an acyl transferase will move the growing molecule on to the next one.
Multi-enzyme complexes are of course more difficult to synthesise and maintain as they are agglomerates of multiple individual enzymes. Thus, these complexes will typically only be assemblied where it is required, where an intermediate cannot be liberated or where a liberated intermediate cannot be sufficiently re-recognised.