You are making two mistakes:

1. Suppressed premises. You are assuming that your scenario is a stochastic process with a non-zero probability of failure on each trial. It doesn't have to be. There is no law of logic or nature that says that it has to be this way. It's just an assumption. It could be, for example, that your footballer cannot ever miss more than twice in a row. It's a more complicated model, to be sure, but it's a possible model.

2. You are over-awed by your intuitions. Our intuitions are passable for everyday, familiar occurrences, but when faced with something as incomprehensible to the imagination as infinity, your intuitions are of little help. Trust your reason, not your gut feelings.

I have to admit that your answer was intriguing and the fact that I reply so late is that I wanted to lay on sheet an outlined idea. So I’m going to begin with point 2 of your reply. It is not about my gut feelings. I can accept the idea of a billion tails in a row because nothing limits this possibility, but when it comes to other things, like the example I’ve mentioned, it’s different – if a big motivated player who scores in big competitions and wins titles can’t hit the ball in the back of his yard for ten years in a row simply because probabilities rules sounds nuts and I simply don’t believe it. Not in a bllinfinite! After all it is a paradigm shift and a contradiction; he could be one of the best players in the world, but he cannot touch a ball (contradiction). So it’s not about the talent and hard-working, it is about probabilities (paradigm shift).
Now, you might say (and you did at the point 1) that it might not be a stochastic system at all. Well, that was the intriguing part for me but I’ll talk about it a bit later.
Meanwhile, I’ve been thinking to some systems. E.G.:
1. Tossing a coin – it’s all about probabilities
2. Poker game. Well, you can have the best poker player in the world losing a billion times in a row against a rookie, simple because his adversary had a first hand of full aces a billion times in a row (100% winning hand).
3. Let’s go back to coins. Let’s presume that someone’s tossing a coin infinite, but this time his purpose is to have tails all the time.
In order to achieve this aim he has to develop a throwing technique - e.g. coin spins twice in air and obtain tails -(presuming this is possible like in sports), so he starts training and eventually he becomes a professional at tossing coins. Now, my first query is which of the following statements is right?
a) We say that he’ll improve his chances of being successful by training, meaning he’s getting the chances of hitting tails are getting higher. The player is never going to be perfect, so he’ll eventually reach a limit in his potential. In this case he will evolve from 50% chances at every try to 99,9%. in an infinite throwings, he has no limit in missing.
b) But my opinion is that this is not about the chances anymore and by improving the technique actually translates in having a success rate. Let’s say, in his worst day he can’t miss more than 3/10. Not being perfect though, he’ll miss 1/10. So, if his value oscillates between A (worst shape), B (best shape), implies results between C (7/10 success) and D (9/10 success) - limits in infinite.
My second question would be why a) or why b)?
Back to the intriguing part. After all, if we have just 99% knowledge about the state of the elements of a system, isn’t fair to say 1% is in the hand of chance and that system is stochastic one as in the poker game for example?
I’ll repeat the questions:
1. Improving chances (probabilities) or reach a success rate?
2. What are the arguments for you choice?
3. Aren’t all the systems where we don't have 100% information about the present elements in it a stochastic one?
4. If you choose ,,success rate” and your questions 3’s answer is yes, than how can limits survive in a stochastic system?

NB: I exclude quantum mechanics and probabilities dictated by subatomic randomness.