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I have always been puzzled as to why the Russians almost never use verbs of possession akin to "have" or "own."

Instead of such verbs, the Russians use the preposition у, whose primary or original meaning is "near" or "at", and sometimes additionally use the verb быть ("be") in the appropriate tense: у меня (есть/была/будет) машина (literally "а car is / was / will be near me"), у моей подруги хороший характер, у него много денег, у этой рыбы острые зубы, у меня хорошее настроение, у меня много дел, and so on. Even if a Russian needs to explicitly stress ownership of, for example, an apartment, he will say something like у меня квартира в собственности rather than я имею квартиру or я владею квартирой, even despite that the latter two constructions are grammatically okay.

The very same grammatical construction is used to express proximity: У дворца роскошный парк ("there is a beautiful park near the palace").

I initially thought that avoiding verbs of possession is common to Slavic languages, but I proved to be wrong. Most other Slavs express possession by verbs akin to "have". For example, in situations where a Russian says у меня много дел, a Pole will say mam wiele rzeczy do zrobienia, and a Serb will say имам пуно посла. It would be pretty unusual for a Russian to say имею много дел, but this is precisely what most other Slavs will say. And below are the most common ways to say "how many yachts do you have?" and "I have a plane" in various Slavic languages:

Polish: Ile masz jachtów? Mam samolot.

Czech: Кolik jachet máš? Mám letadlo.

Serbian: Колико јахти имаш? Имам авион.

Russian: Сколько у тебя (есть) яхт? У меня (есть) самолет.

So my question is this: Why do Russians, in contrast to most other Slavs, almost not use verbs of ownership akin to "have"? In other words, what was the main cultural, historical, or mentality-related factor that resulted in such a difference?

I want to read interesting explanations or hypotheses rather than meaningless answers like "that's the way it is" or "that's just how Russian has evolved." After all, I already know that it is the way it is and that it is how Russian has evolved. The question is why.

Any thoughts are very welcome.

First of all, a shameless plug of my earlier answer on why у does not quite mean "near" (but something more akin to the French chez, i.e. a place/household/domain notion used in the abstract.)

Secondly, I have a general impression that languages usually start out not having a verb for "to have", and then some evolve it and some don't. Entire language families (as far as I'm aware) do without it, such as Turkic or Semitic. The reconstructed Proto-Indo-European of Schleicher's fable doesn't have it, and over all the massive updates it received through the years, no-one challenged the part where "sheep that had no wool" was expressed as "sheep to which wool was not".

Since not much is known about Proto-Slavic syntax and idiomatics, it's hard to make the claim that the у меня construction is older than the reflexes of *jьměti/*jьmati in West and South Slavic languages; у меня is also obviously not quite the same as the proposed Proto-Indo-European dative construction ("is to me"). However, what's fairly clear is that this verb started out with a more literal meaning ("to take") and that's pretty much how all Indo-European languages got their "have" verbs: they were all originally verbs for literal taking or holding. If we trace the history of Spanish all the way back to PIE, we can even see it happening twice. First there were the reflexes of the Latin habeo "hold", then Spanish relegated them to a purely auxiliary function and made tener (also "to hold") its "having" verb.

So in the end, apparently it's not how Russian has evolved; it's how other related languages have evolved and Russian stayed a little more conservative: while not keeping the oldest hypothesised construction, it settled on a similar one instead of the more radical and expressive "hold=>have" approach.

'Near' has nothing to do with 'zero copula' (that's what you tried to explain). In modern Russian it is presented in past/future tense and, in some cases, present/can be added in tense:

У меня была машина/у меня (есть) машина/у меня будет машина (I had a care/I have a car/I will have a car)

I'm not convinced that all thoughts and interesting explanations are useful and fruitful. 'Zero copula' and 'zero verbal predicate' in Russian are relatively well-studied. Zero copula wasn't in use until XIV century as well as zero inflection. And it seems that these events are somewhat related - zero declensional inflection indicates nominative case, zero copula indicates present tense.