The problem is your "constant time" operation now spends some time doing work and some time not. When it's not doing work, the CPU can perform work for other p

The problem is your "constant time" operation now spends some time doing work and some time not. When it's not doing work, the CPU can perform work for other processes or threads and this is, at least in principle, measureable. Thus you haven't eliminated the side channel - just changed the way someone can measure it.