No, FHE is not intrinsically vulnerable to any side-channel attacks, by definition; the properties hold no matter how the "executor" executes the operations and

No, FHE is not intrinsically vulnerable to any side-channel attacks, by definition; the properties hold no matter how the "executor" executes the operations and they include the assumption that obviously they can see (or change!) everything that happens during the execution. However, that effectively means that any FHE approach has to be "anti-optimized", always touching all the data that might get touched, always taking all the conditional paths, not just the worst case path as in constant time algorithms, since the execution must return the correct result for any encrypted input data. It's not that you can "see if we can avoid those with constant-time algorithms or something", the basic table stakes (which all FHE methods actually implement) is the constant-time algorithm approach taken to the extreme and then squared. It does have "some" unavoidable performance impact because of that.