Try to double or triple or even x10 the number of dots on your piece of paper and you get "patterns" forming. Repeat the experiment 1 million times and you might find other patterns appearing.
I love thinking about the concept of randomness, too. Not so much from a mathematical perspective though.
When you flip a coin, the air, the way you flip, the way the coin is manufactured, all affect the outcome in ways we can't really calculate. When you roll a 10-sided die or 6-sided one, the manufacturing of it, the way some sides might have more density than others. The way a "10" is grooved means that side is ever so lighter than the side that has a "1". For sure all these affect the outcome, and, therefore, have a say in how random the coinflip or dice throw is. I've seen people manipulate dice throws, coin flips, measuring exactly how much strength to flip the coin, ensuring every throw has the same number of flips.
The way a dealer shuffles. The way a casino card stack is cut, the way the roulette wheels are oiled. The way the metal ball hits when it's thrown on the wheel.
How random is random? It's a lovely question!
I like the way you explain things that affects its random outcomes, but what if we consider randomness of a computer system? All physical tangible things could be impacted by their physical characteristics on how they were created. What about the computer system who uses random generating functions, do they have any basis at all? Are they really random? How come they are programmed to provide random numbers if computers are precise and absolute.
In computer science there are lot of different kind of random number generators, not all of them are suited for gambling.
You can have Perlin noise or fractal noise which can be used in computer graphics like to generate landscape or Marble or in audio synthesis etc but they will have certain distribution that doesnt make them suitable for gambling.
Uniform distribution is white noise but there are other kind of distribution and algorithm that can be suited for some specific purpose.
But you can easily say there will always be a known pattern in anything generated by a computer program, and the séquence can always be reproduced if you know the initial parameters ( seed etc) and the algorithm.