Well, it's not a rule, it's a guideline. Your first sentence is after all grammatically correct. If we used commas to make the part from who to pump a parenthetical clause as thus:
One of his uncles, who had made millions of dollars in the oil industry in the sixties by inventing a new kind of pump, left John a giant inheritance.
Then it isn't even a difficult sentence to parse.
The reason for the guideline is threefold:
With a greater distance between subject and verb there comes a greater chance of introducing an error or ambiguity by e.g. having a noun in the piece between the subject and verb that could be mistaken for the subject. (So while it's not a rule, following it makes us less likely to break a rule by mistake).
It tends to make for snappier and more easily understood sentences.
It tends to make for shorter sentences, which in turn tends to make for snappier and more easily understood sentences. (Favouring short sentences is a guideline in itself).
It's very important to stress the difference between a guideline and a rule. Knowing the rules let's us write English that is correct. Knowing the guidelines can help us write English that is not only correct, but good.
At the same time, breaking the guidelines is often the best approach to a given sentence, paragraph, etc. that results in stronger writing than if it was followed. Breaking the rules is in the "don't try this at home" territory; there are times that it can be done, but they are rarer, require more skill to carry off, and even the greats can sometimes produce questionable results.
Also, the way we might debate rules and guidelines differs too. If someone were to say (as some have done) that there is a rule against splitting an infinitive, then it can be debunked by showing that it is commonly done and produces a clearly understandable expression. If someone were to say (as others have done) that while there is no such rule, there is a guideline to avoid it, then to argue against one must show that the product of ignoring it is as good, or better, than of following it—and that will be subjective at the best of times.
In terms of your question, this distinction is relevant. The rules of grammar are studied not just by those of us who wish to follow them well, but by those of us who want to understand how language works; to know the explanation for why "the cars is parked outside" is not something we would say (that being an example of agreement, specifically agreement in grammatical number because cars is plural and is is singular).
The guidelines are more given to disagreements of personal opinion (can you imagine if Philip Roth were forced to follow the guideline of avoiding long sentences or Ernest Hemmingway forced to abstain from sentence fragments?), often vague in definition (how large a distance between subject and verb do we need before we consider your guideline broken?), often contradictory (the guideline against long sentences conflicts with that of using longer sentences in the middle of a paragraph than at the beginning or end), and harder to detect (how to judge a conscious effort to keep subject near verb from simply not having much to add between them?).
These features in themselves make them less likely to have name, and less likely again to have names that are universal. I personally think of "proximity" when I'm wondering if I have stretched words too far from their companions, but that would certainly not be a good name since "proximity principle" is term for a grammatical rule.*
The stylistic principles we do have names for tend not to be those that help writing be clear and understandable, as those which are used for specific effects. Often these go against the first set of principles, as diacope, anaphora and epistrophe along with many others go against the general guideline of avoiding repetition.
Between it not being a firm rule and it being vague, I think the odds of finding a fully agreed on name are slim. No doubt some people have names for things, as we're all given to naming things and those of us who worry about distance between subject and verb particularly so, but I'd be more surprised if they are widely accepted and used by others.
*Not an uncontroversial one. The proximity principle has us write "more than one bystander was hurt" rather than "more than one bystander were hurt", because we have was/were agree with bystander which is singular, rather than the whole phrase more than one bystander which is plural. However, some would disagree with this one, and some who would agree would disagree in other cases that some people use it. The proximity principle is perhaps more useful in speaking descriptively of what people do write, than prescriptively about what you think people should write.