I could be mistaken, but I believe that illegal aliens are not “...subject to the jurisdiction thereof...” as that phrase is defined. If so, Senator Howard’s comments would be redundant.
Also, legislative history is relevant, though not conclusive evidence, in determining the intent of laws.
The jails in this country are filled with illegal aliens who found out that they were indeed 'subject to the jurisdiction' of our federal or state governments.
I tried that argument once, and got the response to the effect of “If you think illegal aliens are not subject to US jurisdiction, look at the US inmate population”.