There are two parts of an answer to this, because your questions are somewhat divergent:
> why does it make sense that people can "generate" cash by proving some amount of work done? This of course cannot be done with normal cash.
People do generate money when they work, in a sense, because money doesn't have value. Money represents value. To really understand that you need to think about what money is and why it was invented in the first place.
Before the invention of money there was only direct exchange; I do/give something for/to you and you do/give something for/to me in return. But what if you want what I have but I don't want what you have? Or what if we want something from each other but are too far apart to make the exchange directly? Well, we find a third participant who can act as a kind of transfer agent. They could, for instance, have something I want that you don't want and also want something from you. They trade with you first so now you have something from them that you don't want that you can then trade to me for the thing you want, and everyone is happy. This extends to arbitrarily many, dozens or hundreds even, of intermediate steps.
Now it should be easy to recognize two things:
1) Everyone needing to store a bunch of stuff they don't actually want just so they can pass it on to the next person can become a huge burden for everyone. And how do you store labor anyway? You can't. You can only store goods.
2) Organizing dozens of intermediate links is an extremely difficult problem to solve just so you can get what I have.
The first one can be solved by exchanging IOU vouchers instead. The holder of the voucher becomes entitled to the thing that hasn't yet been given or done. Storing those vouchers is trivially easy compared to storing the things. And you can just as easily store vouchers for work that hasn't been done yet as you can for goods that haven't been given yet.
The second one can be solved by saying what if people put their vouchers into a central voucher bank instead of passing all their vouchers around to each other directly, and then the central voucher bank organizes all the intermediate steps for people without people needing to figure out who has the vouchers they need to complete the chain.
And then once you're there, why even use specific IOUs at all? Why not have all the vouchers be generic but you get different amounts of them instead of different kinds that you can then use freely for anything? And that's obviously what money is.
And from there a new thing should become obvious: The money itself doesn't have any intrinsic value. The labor/good behind it does. Money is just a way of representing the value of something you did/produced in a form that can be easily traded for other things. It's the medium of exchange, not the product. And when there are fewer vouchers in the system relative to what's being produced, each voucher becomes worth more (deflation), and vice versa (inflation). And then the government literally prints and destroys vouchers as needed to try to keep a balance. That is a thing that happens. And so what if there can be prolonged time delays between you doing your work and you receiving your vouchers under some systems? Time delays are not inherent, just practical for bookkeeping. And when long time delays are not practical for bookkeeping they become shorter.
> Are cryptocurrencies supposed to be a potential replacement for real life cash? This was my understanding of the motivation behind Bitcoin, at least.
Only as an unrealistic pretense in the current climate. The reality is that a currency needs to be both moderately inflationary and also very stable to be useful as a medium of exchange of goods/services. You never want it to be a better financial decision to hold onto currency forever instead of using it, and you also never want people to randomly wake up destitute. And regardless of whether bitcoin is technically inflationary in the near term, it is not practically inflationary, and it's definitely not stable.