No, because people never know when they are going to get taxed. Besides, you can't ban fractional reserves.in the real world.
In the case above, the fact that people may get taxed at any instant mean that they are effectively permanently in debt to the state, they just don't know by how much.
A pretty horrific situation!
Here's they key point mansoor - inflation cannot be "controlled" like you are intimating, either in our current monetary system, or your hypothetical one above or indeed in any conceivable system.
I strongly, strongly suspect that any fractional reserve system will eventually fail without government backed deposit insurance protection. Once you grant this protection eventually you end up with taxation (anyway) to give value to all the "badly" created private money.
Do you see what I mean?