Why does oogenesis produce only one functional egg insted of four functional eggs?

Thanks for the really good question. I can quite understand your
exasperation, because textbook examples of meiosis routinely fail to mention that the process depicted is not exactly the one that happens in most female animals, including human women.

In women, meiosis I starts in the ovaries, but the process is arrested in the middle of prophase (the first bit of meiosis I). The primary oocytes (as they are called at this stage) don't carry on with meiosis until ordered to do so by the woman's reproductive hormones - a process that takes place anytime from puberty to menopause.

But now a departure from the textbook pictures becomes evident. After the first meiotic division, you don't get two equal cells, as you would in a man. Rather, you get one big cell (the secondary oocyte) and one runt-like no-hoper, called a polar body. Similarly, after meiosis II, the secondary oocyte is again split into one big cell (the ovum, with the potential to make you or me) and another emaciated polar body. The polar body from the first meiotic division may also go into meiosis II and divide (or it may simply give up at meiosis I), but if it does, it merely produces two more polar bodies.

Thus, although four cells are often produced during a woman's oogenesis, you only really get one viable one.

Hope I've answered your question. That deals with the mechanism behind your query, but I do understand I've not yet provided a reason for that mechanism (why on earth would this elaborate process differ in males and females?). If that's more your angle, let me know, and I'll have a bash at it...