In fact, in typical reactor, neutron needs to travel quite a lot before it initiates next fission, if during these travels it encounters control rod it is "lost" and chain reaction slows down.
Neutron needs to travel because it needs to lose energy (or in other words slow down), this is because modern reactors are designed in such way that fast neutrons wouldnt be enough to support chain reaction (to know why read the rest!). This is a design decision -- you could have a reactor working on fast neutrons -- it just wouldn't be controlable by control rods!
To answer most crucial OP question:
Q: If neutrons travel from the nucleus of one atom to the nucleus of a nearby atom to split it and perpetuate the chain reaction, doesn't that take place within the fuel itself, on an atomic scale?
A: Not really due to the fact that you need to slow down the neutrons, neutrons travel macroscopic paths, and during these travels might be lost to control rod.
In typical modern reactors (experimental ones might be different) you do fission by thermal neutrons (thermal means that these neutrons are in thermal equilibrium with the reactor --- that is have the same speed distribution as it should have in working temperature of the reactor --- neutrons produced by fission have much greater speeds). You can have reactor that works on "fast" (non-thermal) neutrons but these are experimental and much harder to control.
Not every interaction between a neutron and uranium nucleous will result in fission, moreover probability of fission depends on neutron energy. Basically the lower the neutron energy is the more likely fission is. See this chart (from wikipedia):
Typical modern reactors
Neutrons produced in fission have high kinetic energy, so before neutron initiates a fission it must lose most of of the energy, so it's free path is quite long (mean free path is length of path that average neutron travels before initiating next fission). Because of that it is improbable that neutron will initiate fission just after it was produced, because it will still have to much energy. Control rods have plenty of ocasions to catch neutrons.
To slow down the neutrons you'll need them to collide with something (like hydrogen atom, uranium atom and so on). However it turns out that when neutrons collide with heavy atoms they tend not to lose energy, they just change direction (this is just basic mechanics not something nuclear related).
Moderator is a material designed to efficiently slow down the neutrons --- this is a material that has a lot of light atoms (water, graphite, helium). Control rods also displace moderator, so neutrons have lower probability of losing enough energy to initiate fission before they escape the reactor.
You have chain reaction if each fission produces exactly 1.0000 neutron that initiates next fission. To "turn off" the chain reaction you don't need to change this value to 0.0000, not even to 0.9000. Actually if you change the value to 0.99 you'll stop the reaction very fast (I couldn't find precise numbers so this might be off a bit). This is due to the fact that average time between each consecutive fission in chain is low. This means that control rods don't need to drastically alter probability of next fission, just a little bit is enough, even some neutrons wouldn't exit fuel capsule, you need to absorb only tiny fraction to slow down (or kill chain reaction).
What control rods are for
Last thing: control rods are designed to control the reactor and keep it in steady state (that is a proper chain reaction). If you need to shut down the reactor (becaue of some emergency) other means are sometimes employed --- but these specific details of these vary. For example, in some reactors you can introduce "poison" that is a material that absorbs neutrons way faster than uranium, but does not fission, and so starves the chain reaction.
Reactors on fast neutrons
OP asked how does atomic bomb work if there is no moderator there. The answer is you can have both atomic bomb and atomic reactor without moderator. Both can have breeding factor equal or greated than one using fast neutrons. Fission crossection (probability that neutron initiates fission) is lower for fast neutrons but it is enough to both have chain reaction and cascade reaction.
Modern reactors are designed this way that they wouldn't work on fast neutrons --- that is: geometry is designed in a such way that to obtain breeding factor equal to 1.0000 you need thermal neutrons.
It us really easy to create atomic bomb --- you just need to amass a lot of material in the same place and it'll blow.
Obtaining chain reaction on fast neutrons is much harder --- mostly because of the reaction speed. Fast neutrons take a lot of time to lose energy, so you have additional time to slow down the reaction. If you use fast neutrons you don't have this additional time so each step of chain reaction is -- in order of magnitudes! -- faster. In this case control rods are not fast enough and wouldn't stop cascade reaction.
People try to create fast neutron reactors, that use much faster control means. One of the design is accelerator driven systems. In this system breeding factor by design is below 1.0 like 0.995 (once again: these numbers are may be wrong but idea behind them is solid) and additional 0.05 comes from an accelerator (or any other neutron source) that injects neutrons into reactor chamber. It turns out that you can control accelerator intensity fast enough to support chain reaction. Such reactors are "hot" research topic in the field (for example because they can work on "burnt" nuclear fuel).