disclosure: I work for an online p2p payment platform.
While you are correct that sending money via the bank tends to be much much easier, it was not historically the area the Bank wanted to be involved in. To be clear, most banks want to keep your money INSIDE their vaults, making it hard for you to take them out. They actually make money on idle funds you have.
And while you can say that it doesn't cost you a cent, it actually does (at least inherently). You must have an account with said bank to be offered online payment options, which means you either have enough funds to avoid paying monthly banking fee's (which of course means you have enough funds so that you are making the bank money), or you are paying a monthly account fee that covers this cost. To say you aren't paying a fee is naive.
The reason why it is such an issue with companies that are not banks?
1. identification is difficult (consider how much information a bank has about you...do you really want Paypal knowing that much as well?).
2. Compliance issues with registered users. With a man of ill-repute sends money from one person to another, with a bank, that person had to register with registered documents. With companies that are not banks, you are limited in what you can ask for, and what you can keep.
3. Compliance issues on who you can send to (you cannot send "anonymous" money generally for national security reasons).
4. Merchant functionality costs money, and is a big money generator for companies.
5. Money generation. Ie. how do these companies make money? Banks already have a source of income. With companies that are not banks, this generally tend to be fee's on sends (which people hate and complain about...but then never notice they are already paying fee's for their banks!), monthly fee's, or integrations.
Not to say that banks aren't opening up now.
Just as a reference, mastercard has just announced their own P2P payment platform, there is obopay, zoompass, hyperwallet, and a dozen other payment platforms out there.