Do wallets always require a private key to spend coins?
Yes, all wallets have always used private keys. They're what prevents other people from simply going and spending your money, they're used to make the authorisation signatures in a bitcoin transaction. Your best bet is to just use Bitcoin Core to load the wallet.dat file rather than importing it into a third party service. 


  all this to say that my old bitcoin qt wallet never asked me for my private key


The private keys are part of the wallet.dat file and are never user facing in Bitcoin Core.