But really the whole theory does break down if you don't allow for reincarnation.
It is important in this discussion to distinguish between 'reincarnation", which refers to some kind of permanent essence, specifically a single entity of being that leaves one body and becomes another, and 'rebirth' which happens precisely because there is no such permanent being. It might seem like a minor distinction, but it is pivotal to understanding karma in the Buddhist context.
We are fortunate to live at a time when a perfect model for karma exists: the content on the page of the computer screen you are looking at.
For example, suppose you have an image, a .jpg file of a cat. It appears as a single thing, a cat. But in fact, the image is made of thousands or even millions of pixels which in turn are produced by the replication of binary code that leaves (copies) from one computer and travels to another over the internet ( To be precise, the code only exists in the mind. It is an electronic signal which travels).
There is, in fact no single image of a cat that goes from computer to computer. You look closely, it is just a lot of pixels. And if you change that binary code slightly, that image will also change. If I upload an image to the internet, send it to you, and then completely destroy my computer, the causes of that image to reoccur on another computer are still there.
If the conditions are right, meaning that if that binary code can 'take root" (be downloaded) onto another computer, that image will once again appear. This is an analogy to how the Buddhist principle of rebirth occurs. If there is some alteration in the code, the image may be distorted. If it appears on another computer, the user can engage in activities to alter that image (Photoshop). Likewise, you can change the results of your actions by changing your actions.
The roadblock to understanding how the conditions for rebirth occur once the physical body ceases cell replication (when we 'die') comes from the common and popular belief that the material and chemicals in the physical brain alone produce the "person" who then ends up using that brain (the brain produces the mind). From this belief, then quite logically, a brain that ceases to function, ceases to produce its own user, cannot produce a user elsewhere. But this again relies on the idea of a permanent self (or to use our previous analogy, of an actual picture of a cat) that has been generated in the first place, which cannot then replicate. but that is not what Buddhism teaches.
Buddhism does not teach that a self has been generated in the first place. In fact, quite the opposite. No actual self has been generated in the first place, but we mistakenly believe that it has. This mistaken belief ("wrong view") is what leads to attachment and suffering. Instead, a composite of uncountable causes (like seemingly endless binary code) arise "situationally", and it is awareness that experiences this composite as a self. Awareness in this context does not mean personal awareness generated by the senses, but rather a pervading context, like three-dimensional space, in which conditions arise. Rebirth is the experience of reoccurring conditions in the context of awareness. Karma is the tendency for reoccurring conditions to resemble their previous causes. The experience of that karma is the arising of mind.