$15,750 had been taken from me! Distributed to the pool! Not one extra bitcoin was given to me for having the miner that did the work.
How could you think you'd get anything more? That's the WHOLE point of pooled mining: getting regular, small payments, instead of gambling for a BIG payout you'd probably never get anyway.
Also, you protest about things not being clear, but according to the first sentence on Eligius FAQ page "Whenever a block is found, the most recent 25 BTC worth of unpaid shares are paid." If all the coins found in the block are sent to payed shares, then the finder gets nothing more. And according to the first sentence in the bitcoin wiki's article about pooled mining: "Pooled mining is a mining approach where multiple generating clients contribute to the generation of a block, and then split the block reward according the contributed processing power."
"Not one extra bitcoin was given to me for having the miner that did the work." -> You do realize that the miner that finds the block did not more nor less "work" than any other miner in the pool. It's pure luck. It doesn't deserve anything more.
It also enables a fair spread of the initial Bitcoin production: the bitcoin economy may only take off if the coins are more or less spread across all the users, instead of a few that hold them all.
It does NOT compare to communism because everybody does have the same merit. Nobody works "harder" that anybody else. Well, you can double your hashrate, then you'll double your reward. But I repeat: The miner that finds the share has not done any more work than the others. It is pure luck.
Or, you could update your settings for requesting shares of a lot higher difficulty. You'd find them less often, on average, but they'd pay that much more. Same average/expectation value, but much more influence of luck, so you could win more. (Well, on some other pool, because the reward system of eligius prevents anybody from earning more that 100% PPS.)
I don't understand why you say that once again... The pooled mining system spreads the money evenly instead of making a few rich "winners". That is the exact opposite of what you're saying. In any case, it is not satoshi's fault beacause mining pools are built by people on top of Bitcoin; they are not part of bitcoin itself.
Everything cryptocurrency related is SEVERELY lacking in clear explanation, even in the WiKis. Then again, I find ANYTHING that even vaguely relates to cryptography is severely lacking in clarity, even with the BEST of efforts.
So are most textbooks on pretty much every bit of human knowledge. It takes time to write good introductory texts for any topic. And Bitcoin is still in it's beginning! But the best way to learn about something is to try it hands on! But you should check the udemy course on bitcoin. It's really not bad and explains a lot. https://www.udemy.com/bitcoin-or-how-i-learned-to-stop-worrying-and-love-crypto/?sl=E0cddzFTKn1uFFkoCkA%3D