Can reversible computations alone be used to create a computer?
You need workspace (registers set to 0) to read the user's input. Once you've read it, and completed the program, the user needs to erase the input and the output in order to restore the workspace. However, if you assume that the user wants to keep a copy of the input and the output, this is possible to do reversibly.

Similarly for the program, to erase the program from the computer there needs to be another copy stored somewhere.

If you don't allow any reversible operations between the user and the computer, and you insist that everything be done reversibly, then you can't load input into the computer, and you have a useless machine.