Exchange vs Bank (who you got?)

Yesterday I instructed Big Bank to send Bitcoin to my Bitcoin wallet on a Well Known Exchange.

Big Bank had done this on another case for me recently, but to my surprise they would NOT send it that same Bitcoin address (my address) that had been previously used by their system.

Turns out Big Bank requires a unique never been used Bitcoin wallet address for them to send to………..ok fine, they like virgins…..so I contact Well Known Exchange to get another Bitcoin address and THEY inform me the Well Known Exchange does not issue random addresses for each customers transaction: you get the one that you get and that’s it.

So I’ve got a Bank that requires a different wallet address for each Bitcoin transaction. Then I got an Exchange that refuses to comply. Both sides were strangely kinda uppity about the deal as well.

Which one is being unreasonable?

Which leads me to asking for suggestions for a free Bitcoin wallet which will generate its own random Bitcoin address each time I use it to exchange Bitcoin?