what prevents from address generation to accidentally use an existing one? How come bitcoin addresses are created in a split second on my wallet app? what promises the address doesn't override a current existing address? I guess the same question applies for a private key...
what promises the address doesn't override a current existing address?


Nothing does, the security comes in a different way. 

There are 2^160 possible addresses. That number is HUGE. It is so big, that the chance of ever generating an address more than once is basically zero. No matter the power of your supercomputer, or the length of time you have, you will just keep generating new unique addresses.