So traditionally, bankers provided a service by allowing people starting out to borrow money (e.g. mortgages, small business loans) from people who had liquid assets available but not in use (e.g. millionaires, pension funds, other people's savings etc). This is an important function because it increases the value of liquid assets that aren't doing something useful (because now they are doing something useful in the economy - i.e. helping people start out).
Investment banking is the same, but in this case those large pots of cash are being "invested" into companies by buying shares of companies that the bank thinks have good ideas, helping them to grow and be useful in society. When the company matures, the company can sell its shares and use the money made to invest into more smaller companies and help them to grow too.
You've absolutely hit the nail on the head when you point out that these two laudible (and necessary) jobs have been corrupted into "modern banking". Day trading provides no benefit to society as a whole because the shares are not being traded with the company but amongst the share-holders. Lending 100% mortgages was stupid, as is the status quo of lending almost nothing. And shorting stocks causes companies to collapse rather than encouraging them to grow.
The standard argument for why bankers get paid so much is that they make a lot of money. If you make $5m for your company today, shouldn't you be allowed to take $50000 home with you?
My response is this: A checkout assistant at Wal-Mart probably takes nearly $5k from customers in an eight hour shift, and yet he earns minimum wage. Why do we not pay him $500 for his brilliant achievement of earning so much for Wal-Mart? Because Wal-Mart would have made the same with anyone else in his place. The amount earned is independent of him and a consequence of Wal-Mart's back-room staff like deliveries and supply-management.
Similarly, a day-trader might have taken $5m today. But that's not because he's super-clever. It's because he put $1bn on "facebook shares go down" today and came up lucky. To be honest - he didn't even choose to put $1bn on that - the computer told him to and he obeyed.
Our bankers do provide a service to the economy, along with lawyers, journalists, doctors and other professions. But they don't provide a service that is 10x more difficult, or 10x more skilled, and therefore they shouldn't be paid 10x as much. Unless and until the bankers start putting their own money on the stock market, they don't deserve to be making so much by putting other people's money at risk.