The HVX outputs either a 720/60p or a 1080/60i signal. All variable frame rates will be encoded within that transport stream. So yes, you will see the effects of the HVX's variable frame rates, but it's carried within the ATSC-compliant broadcast signal.
The JVC is a very, very small subset of what the HVX does. So yes, it does at least what the JVC does, and a whole lot more, including variable frame rates.
If you wanted to go through the effort to capture the uncompressed output signal, you could most likely conform the footage to extract the pulldown and return it to a native 1080/24p or 720/24p workflow.