Interesting assumption, this, that because so much was going on at Ellis Island, there was no illegal immigration. I would tend to intuit the opposite conclusion -- so much was going on at Ellis Island, that there was plenty left over to "sneak in."
No one could sneak in from Ellis Island. They were sent back if they had any illness, any descrepancy of their documents, any thing at all that would cast a doubt on their being a productive person if allowed to enter would get them sent back. The whole reason of Ellis Island was to sift through the undersirables and eliminate them. You couldn't "sneak" off of Ellis Island unless maybe you could transform yourseld into a boat!
It cost a lot of money to book passage on a ship bound for America, then you had to endure weeks on board a boat, most likely 3rd class or steerage, etc. My grandfather came over in 1912 through Canada, then sent for my grandmother and their 3 kids 3 months later after he got a job and saved up the money for their tickets. It was quite an ordeal even in those days (less than 100 years ago), and I don't envy either of them their "adventure." But they did it the right way - legally and above board.
Not all of the Irish "came through Ellis Island". My forebears arrived in Philadelphia and spread to other states from there. They were granted citizenship right on the ship before landing. All they had to do was to "declare allegiance" to their new country and they were given papers upon landing.
This did not mean that they didn't have to take dirty jobs -- couldn't get much dirtier than coal mining and slate cutting. Many of the men who were old enough to bear arms went right into the Union Army. But the records show that they worked those jobs and others on the side and bought property as soon as they could. The women took in wash and worked as baby nurses and ran boarding houses.