Banks can still make loans simply by offering certificates of deposit. This is the above-board way of loaning out people's money -- you make it absolutely clear that taking it out early has a cost, because the money is locked up in (hopefully) profitable ventures.
Would that be less profitable for the banks? Not really -- they would just adjust their prices to compensate, by charging fees on idle money that's instantly redeemable.
And if you let a secondary market for CDs flower, customers can still get good liquidity. Just in a way that's better subject to market discipline.
That doesn't really sound like a better alternative.
What kinds of terms people choose to transact under is their own business— but all the better when we can be more confident those terms are being followed.
I have no idea whether this is a good idea or if it would work as a business.
In the alternative scenario, economists describe loaning from demand deposits (at a higher interest rate) with a phrase like "the bank is selling liquidity".
Fractional Reserve banks enable you to have your money and be loaned out at the same time(thereby stretching the money supply like an elastic rubber band).
Full Reserve banks ONLY loan out money which was specifically deposited to them for the purposes of being able to be loaned out.