To be fair, the performance problem you're talking about should affect latency rather than throughput. If you can batch lots of operations (not all controlled by the same user) then you can do things as fast as you can without leaking (much) information.
HE hides the execution complexity from the system doing the computation, so no, it can't do someone else's computation and just wait in between to avoid leaking information, it's designed so that the computing operation order and quantity is simply independent on the input data, i.e. the worst case complexity, and a valid HE scheme would have mathematical proof that it's impossible for the system to find a way to do it faster than the worst case.
This is still phenomenally slow, of course.