If there is no change in how often or where ID's are checked, then the ID will have no benefit for controlling illegal immigration. Keep in mind any ID checks would have to include detention and deportation of illegals to be effective. So why have a new ID instead of an air travel card or some other ID for the specific purposes that you have mentioned. Obviously there is a bigger agenda than just safe air travel.
The ID will either be just as useless as the current DLs are (which plenty of illegals do quite well without) or it will be part of a system of increasingly intrusive ID checks which will bag a few illegals (while the rest go around), a few deadbeat dads, some DUI with suspended licenses, etc. Against those meager benefits we will have a new national database, better fed tracking of gun purchases, etc, etc. Clearly not worth it.
It will have one major benefit to controlling illegal immigration - namely, getting a job. Illegals will have a harder time getting bogus IDs.