Visa and the banks are the old way of doing things, they are analog. Blockchain technology and Bitcoin are digital.
Years ago when analog signals had no longer been useful and became outdated, it became a law that all signal for television must be broadcast in digital. This was because analog required so much more energy and bandwidth then digital does.
What bitcoin offers is a replacement for all the energy and resources that are poured into money transmitters, banks, credit cards, and even recently wall street. The technology is expensive to run, but not more expensive than all of the above combined.
And to answer the question of transactions per second, the Bitcoin network is endless. The problem isnt how many transactions can you do, the problem is storage. The blockchain currently is 25 GBs of data that must be stored by many nodes. Though not everyone has to have a nod, if many do not and only big companies do, then it can become once again centralized. Even though 25 GBs is not that much, that is with only around 100-200,000 transactions a day in the BTC network. Visa does 2,000 a second alone.
So if the blockchain was to transact 2,000 a second, you are looking at PB hard drives in a year or so. That is the issue to be concerned with. here is a great article for reading on possibly solutions / problems https://en.bitcoin.it/wiki/Scalability